FUNCTIONS 11 MCGRAW-HILL RYERSON PDF
Transcontinental Printing Ltd. Functions 11 School Board. Barrie, ON. Jacqueline Hill Using Transformations to Graph Functions of the Form y af [ k(x d)]. Functions 11 - Ebook download as PDF File .pdf), Text File .txt) or read book Catholic District School Board Barrie, ON Jacqueline Hill K–12 Mathematics. Premium. 29 · Practice Test for Functions and Quadratics. Chapter 2 Transformations of Functions. Functions and Equivalent Algebraic Expressions.
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Where are the asymptotes of this graph? Repeat parts D through F for the absolute value function f x 5 x. Go to the table of values and scroll up and down the table. Explain why the graph is in two parts with a break in the middle. You have used the slope and y-intercept to sketch lines.
Parent functions include. Explain how each of the following helped you determine the domain and range. When you have finished. Use the table x of values to see what happens to y when x is close to 0 and when x is far from 0. What characteristics of the new 1 parent functions f x 5! Does ERR: Explain why this happens. Quadrants L. Which of the other functions is the resulting graph most like?
Using the table of values and the graph. Tech Support G. Describe 3. What do the graphs have in common? What is different about the graphs? Write equations of the asymptotes for the reciprocal function. Sketch the graphs of f x 5 x and g x 5 1 x on the same axes.
On the same axes. Describe how the three graphs are related. Sketch the graphs of f x 5 x 2 and g x 5 x on the same axes.
This gave me the domain for the function. The equation for this function is v d 5!
Using a Graph The pebble falls a total distance of m. I used 0 X So d can take only values that lie in between 0 and Xscl 20 and 0 Y The distance d is 0 m when the pebble first falls off the edge and m when the pebble lands on the ground.
How can you determine the domain and range of the function v d? So the domain is 0 d The speed of the pebble as it falls to the ground is a function of how far it has fallen.
I used the equation as a check. I used set notation to write the domain and range. I evaluated this using the value operation. I knew that the pebble would gain speed until it hit the ground.
I defined them as sets of real numbers. The pebble starts with no velocity. So 0 is the minimum value of the range. The pebble fell off the edge.
When the pebble lands. As the pebble falls. The pebble must be travelling the fastest when it hits the ground. I saw that the graph started at the origin. I found the domain by thinking about all the values that d could have. Using the Function Equation d 5 0 when the pebble begins to fall. The maximum value of the range is v This happens when d 5 I used the function equation to find how fast the pebble was falling when it landed.
So d must take values between 0 and The graph passes the vertical-line test. I noticed that the x-coordinates were all the integers from 23 to 3 and the y-coordinates were all the integers from 22 to 4.
Why did Sally need to think about the possible values for distance fallen before she graphed the function? What properties of the square root function helped David use the given equation to find the domain and range? The function has a maximum value at the vertex The graph is a circle with centre 0. A closed circle means that the endpoint is included.
Range 5 The graph fails the vertical-line test. The graph is a parabola with a maximum value at the vertex. There are only two y-values. This is the equation of a parabola that opens down. Any value of x will work in the equation. Domain 5 5x [ R6 Range 5 5 y [ R6 This is the equation of a straight line that goes on forever in both directions. There are many vertical lines that cross the graph in two places d Domain 5 5x [ R 25 x 56 Range 5 5 y [ R 25 y 56 This is a function.
I used y instead of f x to describe the range. I realized I had to use values less than or equal to 2. The vertex lies halfway in between the zeros. To find the length. I thought about different possible values of x. A x 5 x 24 2 2x b The smallest the width can Area 5 width 3 length I factored out 22 from 24 2 2x to write the function in factored form.
They need fencing on only three sides of the garden because the house forms the last side. Then the length is 24 2 2x m. This is a quadratic function that opens down. Domain 5 5x [ R 0. It has two zeros. The largest the width can approach is 12 m. I subtracted the two widths from The range of a quadratic function depends on the maximum or minimum value and the direction of opening.
A x The area ranges from 0 to 72 m2. They are usually easier to determine from a graph or a table of values. I substituted x 5 6 into the area function to find the y-coordinate of the vertex. The ranges are restricted because the square root sign refers to the positive square root. Since area must be a positive quantity. The x-coordinate of the vertex is 6. The range of a function depends on the equation of the function. The graph depends on the domain and range. Linear functions of the form f x 5 mx 1 b.
Determine the domain and range of the function f x 5 2 x 2 1 2 2 3 by sketching its graph. Identify which of the relations in questions 1 and 2 are functions.
Text Nelson Functions 11 PDF.pdf
State the domain and range of each relation. State the domain and range of the function. Copy and complete the table to show times for completing the marathon at different speeds. Why is it important for this to be so?
A relation is defined by x 2 1 y 2 5 Participants may walk. The route for a marathon is 15 km long. K a Graph the relation. The graph shows how prices for mailing letters in Canada vary with mass. State the domain and range of the function 1 cup 5 mL. Determine the range of each function if the domain is Use a graphing calculator to graph each function and determine the domain Write the domain and range of each function in set notation.
The large square in the diagram has side length 10 units. The ball reaches a A height of 45 m above the ground after 2 s and hits the ground 5 s after being thrown. A farmer has m of fencing to enclose a rectangular area and divide it into T A ball is thrown upward from the roof of a 25 m building. You can draw a square inside another square by placing each vertex of the inner square on one side of the outer square. Determine the domain and range of each function.
What does function notation mean and why is it useful? If the relation is shown in a mapping diagram. Examples 1. If the relation is described by a list of ordered pairs. If more than one arrow goes from an element of the domain on the left to an element of the range on the right.
If they do. The equation x 2 1 y 2 5 25 does not represent a function because there are two values for y when x is any number between 25 and 5. If you have the equation of the relation. Aid How can you determine whether a relation is a function? If you can draw a vertical line that crosses the graph in more than one place. If you have the graph of the relation. For a relation to be a function.
When a relation is a function. If a single x-value produces more than one corresponding y-value. Because graph A goes on forever in both the positive and negative x direction. Study and 3. So x can be any real number greater than or equal to 21 and y can be any real number greater than or equal to 0. Set notation can be used to describe the domain and range of a function. You can express these facts in set notation: Domain 5 5x [ R6. If you have the graph of a function.
The range is the set of output values that correspond to the input values. You can also determine the domain and range from the equation of a function. Aid The domain of a function is the set of input values for which the function is defined.
Function notation is useful because writing f x 5 3 gives more information about the function—you know that the independent variable is x—than writing y 5 3. You can choose meaningful names. Mid-Chapter Review To evaluate f Because this function has a maximum value at the vertex. Determine the domain and range of each relation in question 1. Graph each function and state its domain and range. For those which are. Determine the domain and range for each.
Use numeric and graphical representations to show that x 2 1 y 5 4 is a function but x 2 1 y 2 5 4 is not a function. A farmer has m of fencing to enclose a d 5 y rectangular area and divide it into three sections as shown. A teacher asked her students to think of a number. Lesson 1. Then she asked them to multiply the resulting difference by the number they first thought of. Use the same scale of to on each axis. Is this relation a function? What is the independent variable in table A?
Is the relation in table A a function? Tom wants to express area in terms of cost to see how much of his yard he can pave for different budget amounts. What is the independent variable? How does this table compare with table A?
The relationship in part E is the inverse of the cost function. The company calculates the cost to the customer as a function of the area to be paved. Graph this inverse relation on the same axes as those in part D. Graph f x. Copy and complete table E for Tom. Write the equation for f x that describes the cost as a function of area.
Reflecting L. Compare this equation with the equation of the inverse you found in part I. Place a Mira along the line y 5 x. To reverse these operations. I wrote down the operations on x in the order they were applied. Multiply by 25 and then add 2. Then I worked backward and wrote the inverse operations. 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?
What do you notice about the two graphs? Where do they intersect? Compare the coordinates of points that lie on the graph of the cost function with those which lie on the graph of its inverse.
Use inverse operations on the cost function.
What do you notice? Write the slopes and y-intercepts of the two lines. How are the domain and range of the inverse related to the domain and range of a linear function? How could you use inverse operations to determine the equation of the inverse of a linear function from the equation of the function? Reversing the Operations In the equation f x 5 2 2 5x.
Is the inverse a function? The graph 5 passes the vertical-line test. Interchanging the Variables f x 5 2 2 5x y 5 25x 1 2 x 5 25y 1 2 I wrote the function in y 5 mx 1 b form by putting y in place of f x.
This use of 21 is different from raising values to the power I solved for y by subtracting 2 from both sides and dividing both sides by The inverse is a function.
The function f is the inverse of the function f. The graph of y 5 f 21 x is a straight 1 line with slope 5 2. I knew that if x.
Communication 21 Tip The inverse is linear. I knew the inverse was a line. I drew the line y 5 x and checked that the graphs of g x and g21 x crossed on that line. That gave me the inverse. I noticed that they were the intercepts. Then I switched the coordinates to find the two points 0. The inverse is a function because it passes the vertical-line test. Plot the points for the inverse and draw the line y 5 x to check for symmetry. I wrote the coordinates of the x.
I plotted the two points of g21 x and joined them with a straight line. I checked that the points on one side of the line y 5 x were mirror images of the points on the other side.
The inverse is not a function: The graph fails the vertical-line test at x 5 1. I plotted the points in red. I wrote the coordinates of the points in the graph and then switched the x. This is the beginning of the domain. The deeper mine has a depth of m. I wrote the temperature function with y and x instead of T d and d. I substituted 22 for T in the equation to get the answer. I wrote the inverse in function notation.
Someone planning a geothermal heating system would need this kind of information. I calculated the beginning and end of the range by substituting d 5 0 and d 5 into the equation for T d. The inverse function is used to determine how far down a mine you would have to go to reach a temperature of.
Because I had switched the variables. I knew that y was now distance and x was temperature. The domain of the inverse is the same as the range of the original function. 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? Which of the relations and inverse relations are functions?
Copy the graph of each function and graph its inverse. To reverse this 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. Determine the inverse relation for each set of ordered pairs. Graph each 2.
Determine the inverse of each linear function by reversing the operations. Then solve for y to determine the inverse. For each linear function. Determine whether each pair of functions described in words are inverses. Add 2. Divide by 3. Sketch the graph of each function in questions 5 and 6.
Multiply by 5. Determine the inverse of each linear function by interchanging the variables. Is each inverse linear? Is each inverse a function?
Multiply by 3. Call the function f and let x represent the temperature in degrees Celsius. Use function notation to express this temperature in degrees Fahrenheit. Use function notation to express this temperature in degrees Celsius.
Graph f and f 21 on the same axes. Double the Celsius temperature. The formula for converting a temperature in degrees Celsius into degrees A Fahrenheit is F 5 9C 1 Compare the slopes of the two lines.
For each function. Explain how you can tell that f 21 is also a linear function. In each case. Let the function g be the method for converting centimetres to inches. State the coordinates of any points that are common to both f and f Multiply by 4 and then divide by Explain what parts c and f represent in question Who might use this rule? Use function notation to represent this amount in inches. Use a chart like the one shown to summarize what you have learned about C the inverse of a linear function.
Find three linear functions that are self-inverse. Explain why the ordered pair 2. Use function notation to represent his height in centimetres. Inverse of a Linear Function Methods: Determine the inverse of the inverse of f x 5 3x 1 4. The ordered pair 1. Write the correct equation for the relation in the form y 5 mx 1 b.
He determined that the equation of the line of best fit for some data was y 5 2. Then evaluate. Self-inverse functions are their own inverses. Ali did his homework at school with a graphing calculator. Once he got home. Compare the effect of these transformations with the effect of the same transformations on quadratic functions.
Do transformations of other parent functions behave in the same way as transformations of quadratic functions? Predict what the graphs of y 5 3f x 1 2 and y 5 3f x 2 1 for each of the other parent functions will look like.
When f x 5! Predict what the graphs of y 5 3! Tech Support 1: Anastasia thinks they could make more interesting patterns by applying transformations to other parent functions as well. Sketch and label each curve on the same axes. Verify your predictions with a graphing calculator. Describe the transformations in words. Without using a calculator.
Shelby wonders whether the transformations will have the same effect on the other functions as they do on quadratic functions. Use brackets when entering transformed versions of y 5 D. Sketch y 5 3x 2 1 2 and y 5 3x 2 2 1 without a calculator. Sketch and label each graph. Make labelled sketches and compare them with transformations on quadratic functions as before. Use a graphing calculator to verify your predictions. NEL 1 Graph the parent functions f x 5 x 2.
The graph of the equation y 5 x 2 1 2 1 2 is the graph of a parabola that 2. How do the 4.
The shape of the graph of g x depends on the graph of the parent function g x and on the value of a. Did the transformations have the same effect on the new parent functions as they had on quadratic functions?
What do you know about the graphs of the following equations? How opens up and has its vertex at 1. Reflecting H. The graph of y 5 x 2 opens up and the graph of y 5 2x 2 opens down. How did the graphs for which a.
The graph of y 5 2x 2 is narrower than the graph of y 5 x 2. How did the effect of transformations on parent functions compare with that on quadratic functions? Copy and complete tables of values for y 5 "x and y 5 "2x. She knows that the parent function is f x 5 "x and that the 2p causes a vertical stretch. Are there any invariant points on the graphs? She wonders 1 what transformation is caused by multiplying x by How could you transform the graph of y 5 "x to obtain the graph of y 5 "2x?
Compare the position and shape of the two graphs. In this formula. State the domain and range of each function. Explain how you would use the graph of y 5 f x to sketch the graph of y 5 f kx.
What happens to the 2 point x. The quadratic function f x 5 x 2 is the parent function. Repeat parts A through D for y 5 "x and y 5 "1 x. How could you use the first table to obtain the second? What happens to the point x. Reflecting I. This transformation is called a horizontal compression of factor 1.
I saw that these functions were y 5 x2. Compare the points in the tables of values. Describe the transformation in words. In both cases. Explain why this is a good description. Using a graphing calculator. 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. I plotted these points and joined them to the invariant point 0. I used the invariant point 0. To graph y 5 0. To graph y 5 4x 2. Instead of using an x-value of 61 to get a y-value of 1.
I used symmetry to complete the graph. That makes sense. Then I used symmetry to complete the other half of the graph. So I multiplied the x-coordinates of the points 1. I used the same x-coordinates as before and multiplied by 5.
The point that originally was 1.
I need an x-value of Point 1. I recognized the V shape of the absolute value function. I switched the values of x and 2x. I shifted the graph of y 5 x 3 units left. Determine the equations. The points that were originally I reasoned that since 2x 5 x. For this graph. The functions graphed in red have equations of the form y 5 f kx. Point 2. The red graph is a compressed version of the green graph that had been flipped over the y-axis. The red graph is further away from the asymptotes than the green graph.
The equation is y 5 1 x. I divided the corresponding x-coordinates to find k: Since the stretch scale factor is 4. The equation is y 5 "24x. Each x-coordinate has been divided by The equation is y 5 1. Since the stretch scale factor is 6. The red graph is the green graph stretched horizontally by the factor 4. The x-coordinates of points on the red graph are 4 times the ones on the green graph.
So I could complete the equation. I applied the horizontal stretch. I multiplied the x-coordinates by 10 to find points on the horizontally stretched graph: Since 0. I multiplied the y-coordinates by 2p. Use transformations to sketch the graph of the pendulum function 1 p L 5 2p" 10 L. Write the equation of the blue graph. The differences in shape are a result of stretching or compressing in a horizontal direction.
I copied the sketch onto a graph with length L on the x-axis and time p L on the y-axis. Period versus Length for a Pendulum y 15 Time s p L 10 5 0 x 5 10 15 20 25 Length m I drew a correctly labelled graph of the situation. Then sketch both graphs on the same set of axes. Write the equation of the red graph. The point 3. Repeat question 4 for each pair of transformed functions. Sketch graphs of each pair of transformed functions. State the coordinates of the 1 d y 5 f 24x 3 4.
In each graph. Describe the transformations in words and note any invariant points. Determine the equations of the transformed functions graphed in red. 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. An equation h Explain why these equations are the same. If you get two different results. Include examples that show how the transformations vary with the value of k.
When an object is dropped from a height. The function y 5 f x has been transformed to y 5 f kx. A quadratic function has equation f x 5 x2 2 x 2 6. How are the transformations alike? How are they different? Extending 1 Determine the T x-intercepts for each function. Suppose you are asked to graph y 5 f 2x 1 4.
Vertical stretch by a factor of 3 The function is a transformed square root function. Apply combinations of transformations. State the domain and range of the transformed function. She would do both stretches or compressions and any reflections to the parent function first and then both translations.
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!
Translate the graph 4 units left and 1 unit down. How do the numbers in the function f x 5 23! 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? Sarit says that she can graph the function in two steps. Do you think this will work? A horizontal stretch by the factor 3 and a reflection in the y-axis means that k 5 2 1.
The vertical asymptote is x 5 0 and the horizontal asymptote is y 5 0. The asymptotes did not change. I drew in the translated asymptotes first. I labelled the graphs and wrote the equations for the asymptotes. Since all the points moved 5 right. 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. I made a sketch of the stretched and reflected graph before applying the translation. Then I drew the stretched and reflected graph in the new position after the translation. 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. Match each equation to its graph.
The equation really is y 5. The point 1. Graph C is wider than the parent function. Graph C is a parabola. The equations for the asymptotes are x 5 21 and y 5 Graph B matches equation 3.
I checked: The vertex is 4. Graph E matches equation 1. This is a transformation of the graph of y 5 becomes A 7. This is the graph of a square root function that has been flipped over the x-axis. Graph D matches equation 2. The parent square root graph has been compressed horizontally or stretched vertically. Equation 5 is the equation of a parabola with vertex 4. This matches equation 1. Since a. The equation must have a. The parent graph has been reflected in the x-axis.
Graph C matches equation 5. It starts at This is the graph of an absolute value function. Equation 7 has a 5 The parent function has been flipped over the y-axis. It has been stretched horizontally. Graph G matches equation 7.
Graph G is a parabola that opens down. This order is like the order of operations for numerical expressions. This is another square root function. Complete the table for the point 1. Add 2 to the x-coordinate. 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 2 13 x x x x21 1 1 1 d y 5 x. Add 4 to the y-coordinate. Explain what transformations you would need to apply to the graph of K y 5 f x to graph each function.
Divide the x-coordinates by 3. Match each operation to one of the transformations from question 1. Multiply the y-coordinates by 5. Multiply the x-coordinates by Sketch each set of functions on the same set of axes. Draw the graph of the new function and write its equation. Use transformations to sketch both graphs. Tomorrow Bhavesh plans to kayak 20 km across a calm lake. Describe the transformations that you would apply to the graph of f x 5 to transform it into each of these graphs.
The graph of g x 5 "x is reflected across the y-axis. Low and high blood pressure can both be dangerous. Doctors use a special index. The next day. Bhavesh uses the relationship Time 5 Speed to plan his kayaking trips. Write the equation of the new function in terms of f. He will need the graph of T s 5 s 2 3 to plan this trip. Assume that normal systolic blood pressure is mm Hg.
He wants to 20 graph the relation T s 5 s to see how the time. Sketch the graph of this index. Distance In the equation Pd 5 P2P. The graph of y 5 f x is reflected in the y-axis. List the steps you would take to sketch the graph of a function of the form 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. 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. 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!
Whenever k is negative. Examples 1 and 2. Study Aid A: How do you apply a horizontal stretch. Then switch x and y and solve for y. Multiply by 3 and then subtract 4. Aid How can you determine the inverse of a linear function? Example 1. The inverse of a linear function is another linear function. This means that you can find the equation of the inverse by reversing the operations on x. It undoes what the original has done. When k is a number greater than 1 or less than 1 The graph of y 5 f kx is the graph of y 5 f x after a horizontal stretch.
When k is a k number between 21 and 1. The inverse of a linear function is the reverse of the original function. When k is negative. 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.
How do you sketch the graph of y have the graph of y 5 f x? Using the functions listed as examples. A ball is thrown upward from the roof of a building 2. Graph each relation and determine which are functions. Use a different method for each function. For each relation. What rule can you use to determine. Principles of Mathematics Principles of Mathematics 9. Nelson Chemistry Fundamentals of International Business: A Canadian Perspective.
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McGraw-Hill Ryerson functions 11 : teacher's resource/
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Reflecting I. Determine where each function is undefined. Point 1.
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