Nelson Functions pdf - Ebook download as PDF File .pdf), Text File .txt) or read book General Manager, Mathematics, Contributing Editors Interior Design. Grade 11 University Math (MCR3U(G)) NEW NELSON TEXTBOOK). MCR3U course outline - New Nelson pdf · MCR3U Assessment Plan. Every student resource page provided in electronic PDF file on CD-ROM for use on whiteboards and projectors. Mathematics Workplace & Everyday Life.

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Publisher, Mathematics. Colin Garnham. Associate Publisher, Mathematics. Sandra McTavish. Managing Editor, Mathematics. David Spiegel. Product Manager. No preview is available for Text Nelson Functions 11 caite.info because its size exceeds MB. To view it, click the "Download" tab above. INVESTIGATE the Math Ang recorded the heights and shoe sizes of students in his class. B. 4 Chapter 1 Height (cm) Shoe Size Height (cm) 10 8

As a result. The graph of y 5 x 2 opens up and the graph of y 5 2x 2 opens down. State any restrictions on the variables. At Phoenix Fashions. How do you add, subtract, and multiply polynomials? Choose one of the parent functions and investigate.

Copy and complete table A. Use the same scale of to on each axis. Graph f x. Write the equation for f x that describes the cost as a function of area.

Graph this inverse of a function inverse relation on the same axes as those in part D. Is the relation in table A a function? What relation can Tom use. Place a Mira along the line y 5 x. What do you notice about the two graphs? Where do they intersect? Reversing the Operations In the equation f x 5 2 2 5x. Write the slopes and y-intercepts of the two lines. Multiply x in the order they were applied. How are the domain and range of the inverse related to the domain and range of a linear function?

Reflecting L. Is the inverse a function? Compare this equation with the equation of the inverse you found in part I. Draw the line y 5 x on your graph. How would a table of values for a linear function help you determine the inverse of that function?

To reverse these operations. What do you notice? Use inverse operations on the cost function. Compare the coordinates of points that lie on the graph of the cost function with those which lie on the graph of its inverse.

How could you use inverse operations to determine the equation of the inverse of a linear function from the equation of the function? The graph of y 5 f 21 x is a straight The inverse is a function. The function f is the inverse 21 1 2 of the function f. This use of f 21 x 5 2 x 1 5 5 21 is different from raising values to the power Interchanging the Variables f x 5 2 2 5x I wrote the function in y 5 mx 1 b form by putting y in place of f x.

The graph 5 passes the vertical-line test. The inverse is linear. Plot the points for the inverse and I plotted the points in red. The inverse is a function because it passes the vertical-line test. I noticed that they g —1 x were the intercepts.

I drew the line y 5 x and checked that the graphs of g x and g21 x crossed on that line. The There are three red points for x 5 1. The inverse is a function. The deeper mine has a depth of m. I substituted 22 for T in the equation to get d 22 5 the answer. I wrote the inverse in 0. Range 5 I calculated the beginning and end of the range by 5T d [ R 11 T d substituting d 5 0 and d 5 into the equation for T d. Someone planning a geothermal heating system would need this kind of information.

This is the beginning of the domain. I knew that y was now d T 5 is the inverse function. It undoes what the original has done and can be found using the inverse operations of the original function in reverse order.

For each graph. This implies that the domain of f is the range of f21 and the range of f is the domain of f Which inverse relations are functions? Determine the inverse relation for each set of ordered pairs. To reverse this function.

Graph each relation and its inverse. Which of the relations and inverse relations are functions? Copy the graph of each function and graph its inverse. Determine the inverse of each linear function by interchanging the variables. Divide by 3. Determine whether each pair of functions described in words are inverses. Sketch the graph of each function in questions 5 and 6. Is each inverse linear? Is each inverse a function? Add 2. For each linear function.

Determine the inverse of each linear function by reversing the operations. Multiply by 5. Then solve for y to determine the inverse. Multiply by 3. Double the Celsius temperature. Call the function f and let x represent the temperature in degrees Celsius. Who might use this rule? In each case. Multiply by 4 and then divide by For each function. Use function notation to express this temperature in degrees Fahrenheit. Let the function g be the method for converting centimetres to inches.

Explain what parts c and f represent in question Use function notation to express this temperature in degrees Celsius. The formula for converting a temperature in degrees Celsius into degrees A Fahrenheit is F 5 95C 1 Once he got home. Use a chart like the one shown to summarize what you have learned about C the inverse of a linear function.

Determine the inverse of the inverse of f x 5 3x 1 4. Then evaluate. Self-inverse functions are their own inverses. Explain why the ordered pair 2. Ali did his homework at school with a graphing calculator. Find three linear functions that are self-inverse.

He determined that the equation of the line of best fit for some data was y 5 2. Use function notation to represent his height in centimetres. Use function notation to represent this amount in inches. Inverse of a Linear Examples: Function Properties: Write the correct equation for the relation in the form y 5 mx 1 b. The ordered pair 1.

Repeat part C for y 5. Describe the transformations in words. Do transformations of other parent functions behave in the same way as transformations of quadratic functions? Tech Support C. Sketch and label each graph. Shelby wonders whether the When f x 5 x. Predict what the graphs of y 5 3f x 1 2 and To enter f x 5 x. Sketch and label 1 transformed versions of y 5: When f x 5! Sketch y 5 3x 2 1 2 and y 5 3x 2 2 1 without a calculator.

Graph the parent functions f x 5 x 2. Predict what the graphs of y 5 3! Use a graphing calculator to verify your predictions. Make labelled sketches and compare them with transformations on quadratic functions as before. Without using a calculator. How did the effect of transformations on parent functions compare with that on quadratic functions?

Examine your sketches for each type of transformation. The graph of y 5 x 2 opens up and the graph of y 5 2x 2 opens down. Did the transformations have the same effect on the new parent functions as they had on quadratic functions? Experiment with each of the parent functions to create patterns on a graphing calculator screen.

Reflecting H. The graph of y 5 2x 2 is narrower than the graph of y 5 x 2. The graph of the equation y 5 x 2 1 2 1 2 is the graph of a parabola that opens up and has its vertex at 1. How do the following graphs compare? What do you know about the graphs of the following equations? The shape of the graph of g x depends on the graph of the parent function g x and on the value of a. How would you compare the graphs of the following pairs of equations?

How did the graphs for which a. Copy and complete tables of values for y 5 "x and y 5 "2x. She wonders what transformation is caused by multiplying x by In this formula. What transformation must be applied to the graph of y 5 f x to get the graph of y 5 f kx? Compare the position and shape of the two graphs.

She knows that the parent function is f x 5 "x and that the 2p causes a vertical stretch. Are there any invariant! What transformation is caused by multiplying L by in the pendulum function p L 5 2p " L? How could you use the first table to obtain the second?

What happens to the point x. Describe the transformation in words. Repeat parts A through D for y 5 "x and y 5 "12 x. Write a summary of the results of your investigations. The I saw that these functions were y 5 x2. Compare the points in the tables of values. In both cases. Explain how you would use the graph of y 5 f x to sketch the graph of y 5 f kx. Reflecting I. Repeat parts A through D for y 5 "x and y 5 "2 x. How is the graph of y 5 f 2x different from the graph of y 5 2f x? What effect does k in y 5 f kx have on the graph of y 5 f x when i k.

Explain why this is a good description. Using a graphing calculator. To graph y 5 0. I used x symmetry to complete the graph. Then I! I knew from the absolute value signs that the parent function was the absolute value function.

Instead of using an x-value of 61 to get a y-value of 1. I knew that the stretch factor was 0. I need an x-value of The point that originally was 1.

That makes sense. To graph y 5 4x 2. So I multiplied the x-coordinates of the points 1. Point 1. The functions graphed in red have equations of the form y 5 f kx. I switched the values of x and 2x. The points 3 that were originally The points that were originally I recognized the V shape of the absolute value function. First I thought about the graph of y 5 2x. For this y graph.

I shifted the graph of y 5 x 3 units! Determine the equations. I reasoned that since 2x 5 x. I divided the corresponding x-coordinates to find k: Each x-coordinate has been multiplied 1 by 6. The red graph is further away from the asymptotes than the green graph. The equation is y 5 14 x. So I could complete the equation. The equation is y 5 1. The green graph has been compressed horizontally The red graph is a compressed version of the green graph and reflected in the y-axis to produce the red graph.

The graph has been stretched horizontally. Point 2. Since the stretch scale factor is 4. The equation is y 5 "24x. The red graph is the green graph stretched The x-coordinates of points on the red graph are 4 times the horizontally by the factor 4.

Since the stretch scale factor is 6. I multiplied the x-coordinates by 10 10 to find points on the horizontally stretched graph: Period versus Length for a Pendulum y 15 Time s p L I drew a correctly labelled graph 10 of the situation. Write the equation of the blue graph. Write the equation of the red graph. I copied the sketch onto a graph with length 5 L on the x-axis and time p L on the y-axis.

The differences in shape are a result of stretching or compressing in a horizontal direction. Repeat question 4 for each pair of transformed functions. In each graph. Describe the transformations in words and note any invariant points. K x and f x 5 x has undergone a transformation of the form f kx. State the coordinates of the image of this point on each graph. The point 3. Sketch graphs of each pair of transformed functions.

Determine the equations of the transformed functions graphed in red. Determine the value of k for each transformation.

Determine the T x-intercepts for each function. If you get two different results.

The function y 5 f x has been transformed to y 5 f kx. How are the transformations alike? How are they different? Extending 1 Explain why these equations are the same. An equation h for this function is t h 5 4. Include examples that show how the transformations vary with the value of k.

For each set of functions. What two transformations are required? Does the order in which you apply these transformations make a difference?

Choose one of the parent functions and investigate. Suppose you are asked to graph y 5 f 2x 1 4. A quadratic function has equation f x 5 x2 2 x 2 6. When an object is dropped from a height. The function is a transformed square root function. Reflection in Horizontal compression the x-axis 1 by a factor of — 2 y 10 First I divided the x-coordinates of points on y 5! How can Neil apply the transformations necessary to sketch the graph? State the domain and range of the transformed function.

How did Neil determine the domain and range of the final function? How does the order in which Neil applied the transformations compare with the order of operations for numerical expressions? I subtracted 4 from each of the x-coordinates and subtracted 1 from each of the y y-coordinates of the graph of y 5 23! I did both shifts together. Sarit says that she can graph the function in two steps.

Do you think this will work? Range 5 5y [ R y Reflecting A. She would do both stretches or compressions and any reflections to the parent function first and then both translations. How do the numbers in the function f x 5 23! The vertical asymptote! I drew in the translated y asymptotes first. The asymptotes did! Range 5 5 y [ R y 2 06 The graphs do not meet their For g x. Graph of g x: To apply the translations.

Since all the points moved 5 right. I used the equations of the asymptotes to help determine the Domain 5 5x [ R x 2 06 domain and range.

I made a sketch of the stretched and reflected graph before applying the translation. This 4 gave me the graph of 2 f! This gave me the graph of 2 y 5 f 5x. The point 1. The equation really is y 5. Graph A is like the graph of y 5.

Graph C is wider than the parent function. Graph B matches equation 3. The equations for the asymptotes are x 5 21 and y 5 The parent graph has been reflected in the x-axis. This matches equation 1. Equation 5 is the equation of a parabola with vertex 4.

The equation must have a. The parent square root graph has been compressed horizontally or stretched vertically. Graph C matches equation 5. Graph D matches equation 2. Graph C is a parabola. This is the graph of a square root function that has been flipped over the x-axis. I checked: The vertex is 4. This is a transformation of the graph of y 5. It starts at This is the graph of an absolute value function. Since a. This order is like the order of operations for numerical expressions.

The parent function has been flipped over the y-axis. Equation 7 has a 5 Graph G matches equation 7. This is another square root function. It has been stretched horizontally. Graph G is a parabola that opens down. Sketch each set of functions on the same set of axes. Complete the table for the point 1. Use words from the list to describe the transformations indicated by the arrows. Explain what transformations you would need to apply to the graph of y 5 f x to graph each function.

A Multiply the y-coordinates by 5. Divide the x-coordinates by 3. Match each operation to one of the transformations from question 1.

Explain what transformations you would need to apply to the graph of K y 5 f x to graph each function. C Add 4 to the y-coordinate. D Add 2 to the x-coordinate. B Multiply the x-coordinates by The next day.

He wants to 20 graph the relation T s 5 s to see how the time. Doctors use a special A index. Assume that normal systolic blood pressure is mm Hg. The graph of g x 5 "x is reflected across the y-axis. In the equation Pd 5 P2P. Tomorrow Bhavesh plans to kayak 20 km across a calm lake.

Distance Draw the graph of the new function and write its equation. Use transformations to sketch both graphs. Sketch the graph of this index. The graph of y 5 f x is reflected in the y-axis. Write the equation of the new function in terms of f. Describe the transformations that you would apply to the graph of f x 5 x to transform it into each of these graphs. Bhavesh uses the relationship Time 5 Speed to plan his kayaking trips.

He will need the graph of T s 5 s 2 3 to plan this trip. Low and high blood pressure can both be dangerous. Match each equation to its graph. Discuss the roles of a. The function y 5 f x has been transformed to y 5 af 3k x 2 d 4 1 c. List the steps you would take to sketch the graph of a function of the form C y 5 af k x 2 d 1 c when f x is one of the parent functions you have studied in this chapter.

How are they alike? Develop a procedure to obtain the graph of g x from the graph of f x. The graphs of y 5 x 2 and another parabola are shown.

Compare the graphs and the domains and ranges of f x 5 x 2 and g x 5! Determine a. Then switch x and y and solve for y. Example 1. The graph of y 5 f kx is the graph of y 5 f x after a horizontal stretch. Multiply by 3 and then subtract 4. When k is a k number between 21 and 1.

It 1 and 2. Examples A1: The inverse of a linear function is the reverse of the original function. How can you determine the inverse of a linear function? If you have the graph of a linear function. How do you apply a horizontal stretch. When k is a number greater than 1 or less than Questions 12 and Whenever k is negative. Then identify all the transformations you need to apply: You can graph the parent function and then apply the transformations 2.

Chapter Review You apply a horizontal compression by dividing the x-coordinates of points y on the original graph by k.

You can graph the function in two steps: Apply both stretches or compressions and any reflections to the parent function first. How do you sketch the graph of y! Using the functions listed as examples. State the domain and range of each function. Sketch the graph of a function whose domain is the set of real numbers and whose range is the set of real numbers less than or equal to 3.

A ball is thrown upward from the roof of a building x 60 m tall. Use a different method for each function. Graph a f x 5 2 x 2 1 2 1 3 each relation and determine which are functions. Explain your i f 3 2 f 2 iii f 1 2 x reasoning. The ball reaches a height of 80 m above! For each relation. Lesson 1. A farmer has m of fencing to enclose a 1. What rule can you use to determine. Three transformations are applied to y 5 x 2: For a fundraising event. Chapter Review Create a function to express the number of Determine the coordinates of the image of this point people as a function of expected income.

Determine the equations of the transformed functions graphed in Explain what the term inverse means in relation to a linear function. Show your steps.

At Phoenix Fashions. How are the domain and range of a linear function related to the domain and range of its inverse? How can you use transformations of parent functions to create other pictures? You must use each parent corresponding domains?

Xscl 1 For each feature. Create your own picture. Explain why you chose each function and each each transformation? Then enter each function listed in the table. List the parts of your picture. The functions used are listed in the table that follows. The first three entries are shown. Begin by putting the calculator in DOT mode. Caroline Herschel. Albert Einstein. Isaac Newton. Nicolaus Copernicus.

Johannes E! Chapter m1m2 Fg! What role does mathematics play in this quest for knowledge? NEL Many great scientists and mathematicians. Polynomial Type Degree The first one has been done for you. For each polynomial in the table at the left. Complete the chart to show what you know about polynomials. Polynomials Examples: How are expanding and factoring related to each other? Use an example in e x2 2 3xy 1 y 2 your explanation. Determine where each function is undefined. Write expressions for the volume and surface area of the new box after the dimensions are doubled.

What are the formulas for the volume and surface area of a rectangular box of width w. Use the given dimensions to write expressions for the volume and surface area of the box. By what factor has the volume increased?

By what factor has the surface area increased? Explain how you know. Suppose each dimension of the box is doubled. By what factors will the surface area and volume of the box increase? I know that with numbers. Are the functions f t and g t equivalent? The altitudes. This distinction Fred simplified f t to g t 5 t 1 Since g t 2 25t 2 The difference in altitude. Simplifying the Polynomial in f t f t 5 25t 2 1 t 1 Polynomials behave like numbers because.

He wants to determine the difference in altitude of two different rockets when their fuel burns out and they begin to coast. What are the advantages and disadvantages of the three methods used to determine whether two polynomials are equivalent? How is subtracting two polynomials like subtracting integers? How is it different? The exception is when the functions intersect.

Evaluating the Functions for the Same Value of the Variable f t 5 25t 2 1 t 1 2 25t 2 1 75t 1 f 0 5 25 0 2 1 0 1 I used t 5 0 because it makes the calculations to find f 0 and g 0 easier. The functions are not equivalent. Y2 Y1 Since the graphs are different. If Fred had not made an error when he simplified.

I zoomed out until I could see both functions. If two functions are equivalent. C1 n 5 25n 1 Both cost functions simplify to the same function. Cheers banquet hall has quoted these charges: Nigel and Petra have created two different functions for the total cost. C2 n 5 10n 1 20n 2 5n 1 1 Are the functions equivalent? C2 n 5 25n 1 The two cost functions are equivalent. The cost of the drinks is 20n 1 The result is two groups of like terms.

The cost of the food is 10n 1 The discount is 5n. The expressions result in different values. The expressions are not equivalent. Evaluating both functions at a single value is sufficient to demonstrate non-equivalence. I substituted The values 0. Show that f x and g x are equivalent by simplifying each. Use two different methods to show that the expressions K 3x 2 2 x 2 5x 2 2 x and 22x 2 2 2x are not equivalent.

Show that f x and g x are not equivalent by evaluating each function at a suitable value of x. Kosuke wrote a mathematics contest consisting of 25 multiple-choice questions. He concluded that two of his functions were equivalent.

The two equal sides of an isosceles triangle each have a length of A 2x 1 3y 2 1. R x 5 x 2 1 x Cost: C x 5 x 1 a Write the simplified form of the profit function. The equations of the functions all appeared to be different.

The perimeter of the triangle is 7x 1 9y. Determine whether each pair of functions is equivalent. Determine two non-equivalent polynomials. Ramy used his graphing calculator to graph three different polynomial C functions on the same axes. Tino owns a small company that produces and sells cellphone cases.

Determine the length of the third side. The scoring system gave 6 points for a correct answer. Kosuke got x correct answers and left y questions unanswered. For each pair of functions. The number 70 can also be expressed as the sum of five consecutive natural numbers: Sanya noticed an interesting property of numbers that form a five-square capital-L pattern on a calendar page: Suppose that f 2 5 g 2.

Express as the sum of five consecutive natural numbers. Show that the functions must be equivalent. Determine the value of the corner number. Express n as the sum of five consecutive natural numbers. Since 70 5 5 3 Kristina reads about an experiment in which a ball is thrown upward from the top of a cliff. I multiplied 5 50t 2 50t 2 25t 1 t 1 25t 1 The height of the ball above the cliff. Kristina learns that the product of the two functions allows her to determine when the ball moves away from.

How can she simplify the expression for v t 3 h t? The distributive property Simplify the expression v t 3 h t 5 t 1 5 25t 2 1 5t 1 2. Is this always necessary? I 5 x 2 1 3x 1 2 x 1 3 multiplied the first two binomials together and got a trinomial. Would Sam have gotten a different answer if he multiplied 25t 2 1 5t 1 2. Reflecting 1. I simplified by 5 x 3 1 6x 2 1 11x 1 6 collecting like terms and arranged the terms in descending order.

I took the 3 term from x 1 3 and multiplied it by the trinomial. Sam grouped together like terms in order to simplify. How does the simplified expression differ from the original? I multiplied directly. I took the 1 term from x 1 1 and multiplied it by the trinomial. I simplified by collecting like 5 x 3 1 5x 2 1 6x 1 x 2 1 5x 1 6 terms and arranged the terms in 5 x 3 1 6x 2 1 11x 1 6 descending order.

Starting with the Last Two Binomials I multiplied the last two binomials V 5 x 1 1 x 1 2 x 1 3 together and got a trinomial. Expanding and Simplifying I wrote the left side of the equation as the product of two identical factors. The simplified expression does not result in 4x 2 1 9y 2 1 16z 2. Using Substitution and then Evaluating I substituted 1 for each of the variables in each Let x 5 y 5 z 5 1 expression to see if the results would be different.

I took the difference of the change in area 5 A2 2 A1 new area. The product gives the area of the new rectangle. Substituting any positive value for w into 2w 2 1 results in a negative number. This means that the new rectangle must have a smaller area than the original one. I w A1 predict that there will be no change in the area. Predict how the area will change if the length of the rectangle is increased by 1 and the width is decreased by 1.

I let w 2w A1 5 2w w represent the width and 2w the length. Write an expression for the change in area and interpret the result. To check my prediction. Their product 5 2w 2 gives the original area.

For any polynomials a. Expand and simplify. Justify your decision. Recall that the associative property of multiplication states that ab c 5 a bc. The kinetic energy of an object is given by E 5 2mv 2.

Determine whether each pair of expressions is equivalent. The two sides of the right triangle shown at the left have lengths x and y. T some. Write a simplified expression for the kinetic energy of the object if a its mass is increased by x b its speed is increased by y 96 Chapter 2 NEL. K a Verify this property for the product 19 5x 1 7 3x 2 2 by expanding and simplifying in two different ways. A cylinder with a top and bottom has radius 2x 1 1 and height 2x 2 1. Explain and illustrate with an example.

Suppose a 3 3 3 3 3 cube is painted red and then divided into twenty-seven 1 3 1 3 1 cubes. Many tricks in mental arithmetic involve algebra. For instance, Cynthie claims to have an easy method for squaring any two-digit number whose last digit is 5; for example, Here are her steps: The numbers 6, 8, 10 also work, since each number is just twice the corresponding number in the b2 example 3, 4, 5.

Show that 5, 12, 13 and 8, 15, 17 are Pythagorean triples. Use the relationship in question 2 to produce three new Pythagorean triples. This relationship was known to the Babylonians about years ago! Use the relationship in question 4 to identify two more new Pythagorean triples.

Testing values of n to determine a pattern. I factored each group by dividing 5 n 2 n 1 3 1 2 n 1 3 by its common factor. Then I factored by dividing each term 5 n 1 3 n 2 1 2 by the common factor n 1 3. Both factors produce numbers greater than 1, so f n can never be expressed as the product of 1 and itself. Quadratic expressions. I can 5 x 1 5 x 2 6 factor it by finding two numbers whose sum is 21 and whose product is These numbers are 5 and I checked the answer by multiplying Check: This left a 5 2 9x 2 2 25 difference of squares.

I used decomposition by finding two 5 10x 2 1 5x 2 6x 2 3 numbers whose sum is 21 and whose product is 10 23 5 I factored the group consisting of the 5 5x 2x 1 1 2 3 2x 1 1 first two terms and the group consisting of the last two terms by dividing each group by its common factor. I divided out the common factor of 5 2x 1 1 5x 2 3 2x 1 1 from each term. I noticed that the first and last terms d 9x 2 1 30x 1 25 are perfect squares.

The square roots 5 3x 1 5 2 are 3x and 5, respectively.

The middle term is double the product of the two square roots, 2 3x 5 5 30x. So this trinomial is a perfect square, namely, the square of a binomial.

I tried to come up with two integers whose sum is 1 and whose product is 6. There were no such integers, so the trinomial cannot be factored. Then I factored the greatest 5 x 3 1 x 2 1 x 1 1 common factor GCF from 5 x 2 x 1 1 1 x 1 1 each pair. Grouping as a difference of squares. Tech Support Use brackets when entering transformed versions of y 5 Without using a calculator. When f x 5! Verify your predictions with a graphing calculator.

Sketch y 5 3x 2 1 2 and y 5 3x 2 2 1 without a calculator. Compare the effect of these transformations with the effect of the same transformations on quadratic functions. Experiment with each of the parent functions to create patterns on a graphing calculator screen. The graph of y 5 2x 2 is narrower than the graph of y 5 x 2. The shape of the graph of g x depends on the graph of the parent function g x and on the value of a.

What do you know about the graphs of the following equations? Examine your sketches for each type of transformation. Reflecting H. The graph of the equation y 5 x 2 1 2 1 2 is the graph of a parabola that opens up and has its vertex at 1. Did the transformations have the same effect on the new parent functions as they had on quadratic functions?

How did the effect of transformations on parent functions compare with that on quadratic functions? How do the following graphs compare? How would you compare the graphs of the following pairs of equations? How did the graphs for which a. The graph of y 5 x 2 opens up and the graph of y 5 2x 2 opens down. Compare the position and shape of the two graphs. Are there any invariant points on the graphs?

State the domain and range of each function. How could you transform the graph of y 5 "x to obtain the graph of y 5 "2x?

She wonders what transformation is caused by multiplying x by She knows that the parent function is f x 5 "x and that the 2p causes a vertical stretch.

In this formula. What transformation must be applied to the graph of y 5 f x to get the graph of y 5 f kx? Copy and complete tables of values for y 5 "x and y 5 "2x. Explain why this is a good description.

NEL I saw that these functions were y 5 x2. Reflecting I. This transformation is called a horizontal compression of factor What happens to the point x. Repeat parts A through D for y 5 "x and y 5 "12 x. Compare the points in the tables of values. How could you use the first table to obtain the second? What effect does k in y 5 f kx have on the graph of y 5 f x when i k. Write a summary of the results of your investigations.

In both cases. The quadratic function f x 5 x 2 is the parent function. How is the graph of y 5 f 2x different from the graph of y 5 2f x? Describe the transformation in words.

Explain how you would use the graph of y 5 f x to sketch the graph of y 5 f kx. What transformation is caused by multiplying L by in the J. Repeat parts A through D for y 5 "x and y 5 "2 x. Using a graphing calculator. To graph y 5 0. Then I used symmetry to complete the other half of the graph. So I multiplied the x-coordinates of the points 1. To graph y 5 4x 2. I joined these points to the invariant point 0.

Instead of using an x-value of 61 to get a y-value of 1. I knew from the absolute value signs that the parent function was the absolute value function. The point that I knew that the stretch factor was 0. I used symmetry to complete the graph. I plotted these points and joined them to the invariant point 0.

That makes sense. I need an x-value of I used the invariant point 0. Determine the equations. I reasoned that since 2x 5 x. NEL I recognized the V shape of the absolute value function. The points that were originally Using a graph to determine the equation of a transformed function In the graphs shown.

For this graph. Point 1. The functions graphed in red have equations of the form y 5 f kx. I shifted the graph of y 5 x 3 units left. I switched the values of x and 2x. Each x-coordinate has been multiplied 1 by 6. The green graph has been compressed horizontally and reflected in the y-axis to produce the red graph. Since the stretch scale factor is 4. Point 2. The equation is y 5 14 x.

The red graph is further away from the asymptotes than the green graph. Each x-coordinate has been divided by Since the stretch scale factor is 6. So I could complete the equation. The red graph is the green graph stretched horizontally by the factor 4. The equation is y 5 1. The red graph is a compressed version of the green graph that had been flipped over the y-axis. The graph has been stretched horizontally. The x-coordinates of points on the red graph are 4 times the ones on the green graph.

The equation is y 5 "24x. The green graph is the square root function because it begins at 0. I divided the corresponding x-coordinates to find k: The red graph is a stretched-out version of the green graph.

Since 0. The original equation was in the form y 5 af kx. I multiplied the y-coordinates by 2p. Period versus Length for a Pendulum y Time s p L 15 10 5 0 x 5 I drew a correctly labelled graph of the situation.

Write the equation of the blue graph. I copied the sketch onto a graph with length L on the x-axis and time p L on the y-axis. Write the equation of the red graph. Then sketch both graphs on the same set of axes. The differences in shape are a result of stretching or compressing in a horizontal direction. Describe the transformations in words and note any invariant points. Sketch graphs of each pair of transformed functions.

Repeat question 4 for each pair of transformed functions. In each graph. Determine the equations of the transformed functions graphed in red.

State the coordinates of the image of this point on each graph. The point 3. What two transformations are required? Does the order in which you apply these transformations make a difference?

Choose one of the parent functions and investigate. An equation h for this function is t h 5 4. The function y 5 f x has been transformed to y 5 f kx. Include examples that show how the transformations vary with the value of k.

Determine the T x-intercepts for each function. If you get two different results. Suppose you are asked to graph y 5 f 2x 1 4. A quadratic function has equation f x 5 x2 2 x 2 6. Explain why these equations are the same. When an object is dropped from a height. For each set of functions. Determine the value of k for each transformation. How are the transformations alike? How are they different?

Extending 1 Horizontal compression 1 by a factor of — 2 10 y First I divided the x-coordinates of points on y 5! How can Neil apply the transformations necessary to sketch the graph? Vertical stretch by a factor of 3 The function is a transformed square root function. State the domain and range of the transformed function.

How do the numbers in the function f x 5 23! How did Neil determine the domain and range of the final function? She would do both stretches or compressions and any reflections to the parent function first and then both translations. Range 5 5y [ R y Reflecting A. Do you think this will work? How does the order in which Neil applied the transformations compare with the order of operations for numerical expressions? Sarit says that she can graph the function in two steps.

I subtracted 4 from each of the x-coordinates and subtracted 1 from each of the y-coordinates of the graph of y 5 23! The vertical asymptote is x 5 0 and the horizontal asymptote is y 5 0. A horizontal stretch by the factor 3 and a reflection in the y-axis means that k 5 2 Graph of g x: I drew in the translated asymptotes first. I labelled the graphs and wrote the equations for the asymptotes. To apply the translations.

Since all the points moved up 4. The graphs do not meet their asymptotes. I used the equations of the asymptotes to help determine the domain and range. The asymptotes did not change.

The graph of y 5 f 25x looked the same because the y-axis is the axis of symmetry for y 5 f 5x. This gave me the graph of y 5 f 5x. The point 1. NEL 1. The equation really is y 5. Graph B matches equation 3. This is a transformation of the graph of y 5 1.

Equation 5 is the equation of a parabola with vertex 4. Graph C is a parabola. The parent graph has been reflected in the x-axis. The equations for the asymptotes are x 5 21 and y 5 Graph C matches equation 5. It starts at Since a. Graph E matches equation 1. The equation must have a. This is the graph of an absolute value function. The parent square root graph has been compressed horizontally or stretched vertically.

Graph C is wider than the parent function. This is the graph of a square root function that has been flipped over the x-axis. I checked: The vertex is 4.

This matches equation 1. It has been stretched horizontally. Graph G is a parabola that opens down. The parent function has been flipped over the y-axis. Graph G matches equation 7. This is another square root function. Equation 7 has a 5 This order is like the order of operations for numerical expressions. Add 2 to the x-coordinate.

Add 4 to the y-coordinate. Multiply the y-coordinates by 5. Complete the table for the point 1. Explain what transformations you would need to apply to the graph of K y 5 f x to graph each function. Multiply the x-coordinates by Explain what transformations you would need to apply to the graph of y 5 f x to graph each function.

Divide the x-coordinates by 3. Sketch each set of functions on the same set of axes. Match each operation to one of the transformations from question 1. Use words from the list to describe the transformations indicated by the arrows.

Draw the graph of the new function and write its equation. Sketch the graph of this index. Tomorrow Bhavesh plans to kayak 20 km across a calm lake. Bhavesh uses the relationship Time 5 Speed to plan his kayaking trips.

Assume that normal systolic blood pressure is mm Hg. Use transformations to sketch both graphs. Low and high blood pressure can both be dangerous. The next day. He will need the graph of T s 5 s 2 3 to plan this trip. Doctors use a special A index. In the equation Pd 5 P2P. The graph of y 5 f x is reflected in the y-axis.

Describe the transformations that you would apply to the graph of f x 5 to transform it into each of these graphs. Distance Write the equation of the new function in terms of f. He wants to 20 graph the relation T s 5 s to see how the time. The graph of g x 5 "x is reflected across the y-axis.

Match each equation to its graph. Discuss the roles of a. How are they alike? Develop a procedure to obtain the graph of g x from the graph of f x. Determine a. The function y 5 f x has been transformed to y 5 af 3k x 2 d 4 1 c. Compare the graphs and the domains and ranges of f x 5 x 2 and g x 5! List the steps you would take to sketch the graph of a function of the form C y 5 af k x 2 d 1 c when f x is one of the parent functions you have studied in this chapter.

The graphs of y 5 x 2 and another parabola are shown. It undoes what the original has done. Questions 12 and Examples Q: How can you determine the inverse of a linear function?

The inverse of a linear function is the reverse of the original function. When k is a k number between 21 and 1. Chapter 1 21 x 5 x Input Inverse A2: The graph of y 5 f kx is the graph of y 5 f x after a horizontal stretch. How do you apply a horizontal stretch. When k is a number greater than 1 or less than 1 Then switch x and y and solve for y. Whenever k is negative. Example 1. This means that you can find the equation of the inverse by reversing the operations on x.

The inverse of a linear function is another linear function. Multiply by 3 and then subtract 4. Chapter Review You apply a horizontal compression by dividing the x-coordinates of points on the original graph by k. Then identify all the transformations you need to apply: You can graph the function in two steps: Apply both stretches or compressions and any reflections to the parent function first.

When k is negative. NEL 3.

How do you sketch the graph of y! You can graph the parent function and then apply the transformations one by one. Sketch the graph of a function whose domain is the set of real numbers and whose range is the set of real numbers less than or equal to 3.

A farmer has m of fencing to enclose a rectangular area and divide it into two sections as shown. Graph each relation and determine which are functions. The ball reaches a height of 80 m above the ground after 2 s and hits the ground 6 s after being thrown.

A ball is thrown upward from the roof of a building 2! What rule can you use to determine. Using the functions listed as examples. For each relation.

Use a different method for each function. Three transformations are applied to y 5 x 2: Chapter Review State the domain of this new function. Determine the coordinates of the image of this point on the graph of y 5 3f x 1 1 4 2 2.

Create a function to express the number of people as a function of expected income. For a fundraising event. Show your steps. At Phoenix Fashions. How are the domain and range of a linear function related to the domain and range of its inverse? Explain what the term inverse means in relation to a linear function. You must use each parent x function at least once. Create your own picture. The first three entries are shown. List the parts of your picture.

Explain why you chose each function and each transformation. For each feature. Xscl 1 The functions used are listed in the table that follows. Begin by putting the calculator in DOT mode. Then enter each function listed in the table. Isaac Newton. Caroline Herschel.

Albert Einstein. Johannes Kepler.

Nicolaus Copernicus. Many great scientists and E! What role does mathematics play in this quest for knowledge? Chapter 2 m1m2 Fg! Complete the chart to show what you know about polynomials. Study Aid For help. Determine where each function is undefined. The first one has been done for you.

How are expanding and factoring related to each other? Use an example in x2 2 3xy 1 y 2 your explanation. For each polynomial in the table at the left. What are the formulas for the volume and surface area of a rectangular box of width w.

Use the given dimensions to write expressions for the volume and surface area of the box. By what factor has the volume increased? By what factor has the surface area increased? Explain how you know. Suppose each dimension of the box is doubled. By what factors will the surface area and volume of the box increase? Write expressions for the volume and surface area of the new box after the dimensions are doubled. Simplifying the Polynomial in f t f t 5 25t 2 1 t 1 5 25t 2 1 t 1 1 5t 2 2 75t 2 Polynomials behave like numbers because.

The altitudes. The difference in altitude. He wants to determine the difference in altitude of two different rockets when their fuel burns out and they begin to coast. Are the functions f t and g t equivalent? Selecting a strategy to determine equivalence Determine if f t and g t are equivalent functions. This distinction is necessary because both functions are named with the letter a.

I know that with numbers. The functions are not equivalent. If two functions are equivalent. If Fred had not made an error when he simplified. Evaluating the Functions for the Same Value of the Variable f t 5 25t 2 1 t 1 2 25t 2 1 75t 1 f 0 5 25 0 2 1 0 1 2 25 0 2 1 75 0 1 I used t 5 0 because it makes the calculations to find f 0 and g 0 easier.

I zoomed out until I could see both functions. The exception is when the functions intersect. Y2 Y1 Since the graphs are different. How is subtracting two polynomials like subtracting integers? How is it different? What are the advantages and disadvantages of the three methods used to determine whether two polynomials are equivalent? The discount is 5n. The cost of the drinks is 20n 1 C2 n 5 10n 1 20n 2 5n 1 1 Are the functions equivalent?

Nigel and Petra have created two different functions for the total cost. The two cost functions are equivalent. Cheers banquet hall has quoted these charges: The cost of the food is 10n 1 Both cost functions simplify to the same function. The result is two groups of like terms. I substituted some values for x. The values 0. The expressions result in different values. Evaluating both functions at a single value is sufficient to demonstrate non-equivalence.

The expressions are not equivalent. Use two different methods to show that the expressions K 88 Chapter 2 3x 2 2 x 2 5x 2 2 x and 22x 2 2 2x are not equivalent. Show that f x and g x are not equivalent by evaluating each function at a suitable value of x. Show that f x and g x are equivalent by simplifying each. R x 5 x 2 1 x Cost: C x 5 x 1 a Write the simplified form of the profit function. The scoring system gave 6 points for a correct answer. Ramy used his graphing calculator to graph three different polynomial C NEL functions on the same axes.

Kosuke got x correct answers and left y questions unanswered. He concluded that two of his functions were equivalent. Tino owns a small company that produces and sells cellphone cases. Determine whether each pair of functions is equivalent. Equivalent Algebraic Expressions The two equal sides of an isosceles triangle each have a length of A 2x 1 3y 2 1.

Determine the length of the third side. The perimeter of the triangle is 7x 1 9y. Determine two non-equivalent polynomials. The equations of the functions all appeared to be different. Kosuke wrote a mathematics contest consisting of 25 multiple-choice questions.

For each pair of functions. Suppose that f 2 5 g 2. The number 70 can also be expressed as the sum of five consecutive natural numbers: Show that the functions must be equivalent. Determine the value c of the corner number. Express n as the sum of five consecutive natural numbers. Sanya noticed an interesting property of numbers that form a five-square capital-L pattern on a calendar page: Since 70 5 5 3 Express as the sum of five consecutive natural numbers.

Write expressions for the sum of the five numbers. The height of the ball above the cliff. Kristina learns that the product of the two functions allows her to determine when the ball moves away from.

Kristina reads about an experiment in which a ball is thrown upward from the top of a cliff. I multiplied each of the three terms in the trinomial by each of the terms in the binomial. How can she simplify the expression for v t 3 h t? The distributive property Simplify the expression v t 3 h t 5 t 1 5 25t 2 1 5t 1 2.

Would Sam have gotten a different answer if he multiplied 25t 2 1 5t 1 2. I took the 3 term from x 1 3 and multiplied it by the trinomial. Reflecting 1. I simplified by collecting like terms and arranged the terms in descending order.

Then I took the x-term from x 1 3 and multiplied it by the trinomial. Sam grouped together like terms in order to simplify. How does the simplified expression differ from the original? Is this always necessary?

I multiplied the first two binomials together and got a trinomial. NEL I simplified by collecting like terms. Using Substitution and then Evaluating Let x 5 y 5 z 5 1 L.

Since the left side did not equal the right side. I took the 1 term from x 1 1 and multiplied it by the trinomial. I multiplied directly. Starting with the Last Two Binomials V 5 x 1 1 x 1 2 x 1 3 I multiplied the last two binomials together and got a trinomial.

Expanding and Simplifying 2x 1 3y 1 4z 2 5 2x 1 3y 1 4z 2x 1 3y 1 4z I wrote the left side of the equation as the product of two identical factors. Predict how the area will change if the length of the rectangle is increased by 1 and the width is decreased by 1. Substituting any positive value for w into 2w 2 1 results in a negative number. This means that the new rectangle must have a smaller area than the original one.

Write an expression for the change in area and interpret the result. I took the difference of the new area. I increased the length by 1 and decreased the width by 1 by adding and subtracting 1 to my previous expressions. The product gives the area of the new rectangle. Their product gives the original area.

I predict that there will be no change in the area. I let w represent the width and 2w the length. Justify your decision.

Expand and simplify. For any polynomials a. Recall that the associative property of multiplication states that ab c 5 a bc. A Write a simplified expressions for its a surface area. Write a simplified expression for the kinetic energy of the object if a its mass is increased by x b its speed is increased by y The two sides of the right triangle shown at the left have lengths x and y.

The kinetic energy of an object is given by E 5 NEL. T some. K 2x " 1 2x 1 Verify this property for the product 19 5x 1 7 3x 2 2 by expanding and simplifying in two different ways. A cylinder with a top and bottom has radius 2x 1 1 and height 2x 2 1.

Determine whether each pair of expressions is equivalent. Explain and illustrate with an example. Many tricks in mental arithmetic involve algebra. For instance, Cynthie. Here are her steps: The number that results will be the answer you want.

A Pythagorean triple is three natural numbers that satisfy the equation of the Pythagorean theorem, that is, a 2 1 b 2 5 c 2.

An example of a Pythagorean triple is 3, 4, 5, since 32 1 42 5 The numbers 6, 8, 10 also work, since each number is just twice the corresponding number in the example 3, 4, 5. Show that 5, 12, 13 and 8, 15, 17 are Pythagorean triples.

Use the relationship in question 2 to produce three new Pythagorean triples. This relationship was known to the Babylonians about years ago! Use the relationship in question 4 to identify two more new Pythagorean triples.

After some calculations and guess and check, I found a pattern. The first factor was of the form n 1 3 and the second factor was of the form n 2 1 2. Since both factors produce numbers greater than 1, f n can never be expressed as the product of 1 and itself.

Both factors produce numbers greater than 1, so f n can never be expressed as the product of 1 and itself. This is a trinomial of the form ax 2 1 bx 1 c, where a 5 1. I can factor it by finding two numbers whose sum is 21 and whose product is These numbers are 5 and First I divided each term by the common factor, 2. This left a difference of squares. I took the square root of 9x 2 and 25 to get 3x and 5, respectively.

This is a trinomial of the form ax2 1 bx 1 c, where a 2 1, and it has no common factor. I used decomposition by finding two numbers whose sum is 21 and whose product is 10 23 5 I factored the group consisting of the first two terms and the group consisting of the last two terms by dividing each group by its common factor.

I divided out the common factor of 2x 1 1 from each term. I noticed that the first and last terms are perfect squares. The square roots are 3x and 5, respectively. The middle term is double the product of the two square roots, 2 3x 5 5 30x. So this trinomial is a perfect square, namely, the square of a binomial. Trinomials of this form may be factored by decomposition.

I tried to come up with two integers whose sum is 1 and whose product is 6. There were no such integers, so the trinomial cannot be factored. Then I factored the greatest common factor GCF from each pair. Then I factored out the greatest common factor, x 1 1 , to complete the factoring. So factoring is the opposite of expanding. This is only possible if the grouping of terms allows you to divide out the same common factor from each group. What are the lengths of the sides of the pond?

Saturn is the ringed planet most people think of, but Uranus and Neptune A. In addition, there are ringed planets outside our solar system. Consider the cross-section of the ringed planet shown. Write factored expressions for i the area of the region between the planet and the inner ring ii the area of the region between the planet and the outer ring iii the difference of the areas from parts i and ii b What does the quantity in part iii represent? For each path through the flow chart, give an example of a polynomial that would follow that path and show its factored form.

Explain how your flow chart could describe how to factor or show the non-factorability of any polynomial in this chapter. The first three terms form a perfect square. This is now a difference of squares. Expanding confirms that x 2 2 1 5 x 2 1 x 1 1 and also that.

Make conjectures and determine similar factorings for each expression. The French mathematician Mersenne was interested in finding the values of m that produced prime numbers, n. Expand the expressions that contain powers, treating them like polynomials, to show that you get 2 6 2 1. Using these types of patterns, show that n 5 2 1 is composite.

How can you determine whether two polynomials are equivalent? You can simplify both polynomials. If their simplified versions are the same, the polynomials are equivalent; otherwise, they are not. Simplifying yields f x 5 x 2 2 4x 1 4,. So f x and g x are equivalent, but h x is not equivalent to either of them. If the domains of two functions differ in value for any number in both domains, then the functions are not equivalent. For example, for the functions above, f 0 5 4 while h 0 5 So f x and h x are not equivalent.

You can graph both functions. If the graphs are exactly the same, then the functions are equivalent; otherwise, they are not. When the variables of a polynomial are replaced with numbers, the result is a number. The properties for adding, subtracting, and multiplying polynomials are the same as the properties for the numbers.

For any polynomials a, b, c: Commutative Property. Distributive Property a b 1 c 5 ab 1 ac; a b 2 c 5 ab 2 ac. Because of the distributive property, the product of two polynomials can be found by multiplying each term in one polynomial by each term in the other and can be simplified by collecting like terms.

Justify your answer. The sum of the ages of Pam, Dion, and their three. Pam is five years younger than Dion. What is the sum of the ages of their children? Determine the possible values of k. Define rational functions, and explore methods of simplifying the related rational expression. To start the game, Adonis announces he will draw n numbers from a set that includes all the natural numbers from 1 to 2n. The players then pick three numbers. Adonis draws n numbers and announces them.

The players check for matches. Any player who has at least two matches wins. A rational function can be expressed as R x , where R and S are f x 5 S x polynomials and S 2 0; for example, x 2 2 2x 1 3 1 ,x2 4x 2 1 4 A rational expression is a quotient of polynomials; for example,. For example, if Adonis draws 5 numbers from the set 1 to 10, the probability of winning is P 5 5. The game is played at a rapid pace, and Adonis needs a fast way to determine the range he should use, based on the number of players and their chances of winning.

Write the simplified expression for the function defined by 3n3 2 3n2 P n 5 3. I knew that I could simplify rational numbers by first factoring numerators and denominators and dividing each by the common factor 24 3 8 8 ae. I factored the numerator and denominator of P n. Since I cannot divide by zero, I determined the restrictions by calculating the values of n that make the factored denominator zero.

I solved 4n 2n 2 1 n 2 1 5 0 by setting each factor equal to 0. These are the zeros of the denominator or, equivalently, the numbers that are not in the domain of the function. How is working with rational expressions like working with rational numbers? How do the restrictions on the rational expression in P n relate to the domain of this rational function? This gives the restrictions x. I determined the restrictions by solving 2x 2 5 0 to get the zeros of the original denominator.

I determined the restrictions by finding the zeros of the original denominator by solving 26x7y 5 0.

Then I divided both the numerator and denominator by 2x. The only restriction is x 2 0. Selecting a strategy for simplifying the quotient of a polynomial and a monomial Simplify and state any restrictions on the variables.

Then I divided the numerator and denominator by the GCF. The restrictions are x 2 23y. The only restriction is x 2 1. I determined the restrictions by solving 2 1 2 x 5 0. Selecting a strategy for simplifying the quotient of quadratics in two variables Simplify and state any restrictions on the variables: I divided the numerator and denominator by the GCF 1 2 x.

I divided out the common factor. This means that f x is undefined when x 5 1. These numbers must be excluded or restricted from being possible values for the variables. State any restrictions on the variables. If the denominator contains two or more terms.

As a result. State the domain of each function. An isosceles triangle has two sides of length 9x 1 3. Determine the ratio of their volumes. Explain how you found each answer.