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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Present Value 8. Identify a function as a special type of relation Recognize functions in various representations and use function notation Explore the properties of some basic functions and apply transformations to those functions Investigate the inverse of a linear function and its properties.
For help, see the Essential Skills Appendix. Then sketch the graph. Solve each quadratic equation. Begin by completing a table like the one shown. Then list similarities and differences among the three types of relations. Property Equation s Shape of graph Number of quadrants graph enters Descriptive features of graph Types of problems modelled by the relation Linear Relations Circles Quadratic Relations.
The creek that goes through her farmland will form one side of the rectangular boundary. Rebecca considers different widths to maximize the area enclosed.
What are the minimum and maximum values of the width of the field? What equations describe each? Width m Length m Area m2. Graph the data you wrote in the first two columns.
Use width as the independent variable. Describe the graph. What type of relationship is this? Now graph the data you wrote in the first and third columns. Use width as the independent variable again.
How could you have used the table of values to determine the types of relationships you reported in parts D and E? How could you have used the equations from part B to determine the types of relationships you reported in parts D and E?
Describe the relationship shown in the scatter plot. Use your plot to predict the height of a person with each shoe size. Write the equation of your line. Shoe Size 10 Plot the data. How are your results different from those in part B? Is the relation drawn in part D a function? Which of the relations in parts A and D could be used to predict a single height for a given shoe size?
I wrote the relation as a set of ordered pairs. For example. How did the numbers in the table of values show that the relation was not a function? How did the graph of the linear function you drew in part D differ from the graph of the relation you plotted in part A? Explain why it is easier to use the linear function than the scatter plot of the actual data to predict height. I listed the day for practice—the dependent variable. For each of the given relations.
Explain why the relation plotted in part A is not a function. Each element of the domain corresponds with only one element in the range. I drew another mapping diagram for the age and practice day relation. This cannot be a function. Each element of the domain has only one corresponding element in the range. I can predict his or her practice day.
In this case. The diagram is called a mapping diagram. Using a Mapping Diagram a Student Craig Magda Stefani Amit Practice day Sat Thurs Tues I drew a diagram of the relation between students and soccer practice days by listing the student names in an oval and the days in another oval. The elements in the right oval are the values of the dependent variable and make up the range.
I wrote the domain and range by listing what was in each oval. This is not a function. This relation is not a function. I matched the ages to the practice days. Two arrows go from 15 to two different days. The relation is a function because each student name has only one arrow leaving it. Then I drew arrows to match the students with their practice days. The first elements appear only once in the list of ordered pairs.
No name is repeated. I noticed that one year-old practiced on Tuesday. Each student has only one practice day. This is not the graph of a function. This showed that each x-value in the domain corresponds with only one y-value in the range.
The ruler crossed the graph in two places everywhere except at the leftmost and rightmost ends of the circle. This is the graph of a function. This showed that there are x-values in the domain of this relation that correspond to two y-values in the range. Wherever I placed my ruler. Any vertical line drawn on the graph intersects the graph at only one point. NEL Introduction to Functions 7. I used my graphing calculator and entered the upper half of the circle in Y1 and the lower half in Y2.
This graph fails the vertical-line test. This graph passes the vertical-line test. Its graph is a straight line that increases from left to right. Using the Graph Defined by its Equation a This equation defines the graph of a linear function I used my graphing calculator and entered y 5 2x 2 5. I used my graphing calculator to enter y 5 2x 2 2 3x 1 1 and applied the vertical-line test to check. Then I applied the vertical-line test to check. I graphed the function and checked it with the vertical-line test.
Substituting Values a For any value of x. No matter what number I substituted for x. I get only one number for y that satisfies the equation. No matter what number I choose for x. Linear relations. This shows that there is only one element in the range for each element of the domain. A graph represents a function if every vertical line intersects the graph in at most one point.
I got two values for y with x 0. If you can find even one value of x that gives more than one value of y when you substitute x into the equation.
I got only one answer for y when I doubled the number for x and then subtracted 5. Vertical lines are not functions but horizontal lines are. This is usually represented by the y-values on a coordinate grid.
This is usually represented by the x-values on a coordinate grid. Quadratic relations. NEL Introduction to Functions 9. This equation represents a function. Use a ruler and the vertical-line test to determine which graphs are functions. Substitute 26 for x in each equation and solve for y. State which relations are functions.
Use your results to explain why y 5 x 2 2 5x is a function but x 5 y 2 2 5y is not. Graph the relations in question 4. Explain your reasoning. Determine which of the following relations are functions.
The grades and numbers of credits for students are listed. Describe the graphs of the relations y 5 3 and x 5 3. Then graph each function and use the vertical-line test to determine whether your prediction was correct. Then use the vertical-line test to confirm your answers to part c. Are these relations functions? Repeat for x 5 Identify each type of relation and predict whether it is a function. Use numeric and graphical representations to investigate whether the relation a y 5!
Determine which relations are functions. The cost of renting a car depends on the daily rental charge and the number of kilometres driven. Support your answer. A graph of cost versus the distance driven over a one-day period is shown. Function Characteristics: Use a chart like the following to summarize what you have learned about C Examples: Extending What is its range?
When the fox reaches point F. The rabbit sees the fox and runs in a straight line to its burrow. The resulting curve is called a curve of pursuit. How do they differ from the one in question 1? NEL Introduction to Functions The fox continuously adjusts its direction so that it is always running directly toward the rabbit.
The fox then changes direction to run along line EA. What is the temperature at the bottom of each mine? Another South African mine. I wrote the equation again. East Rand mine in South Africa. Since this equation represents a linear relationship between temperature and depth. Using an Equation a An equation for temperature is Communication Tip The notations y and f x are interchangeable in the equation or graph of a function.
Western Deep. I wrote a linear equation for the problem. Temperature is a function of depth. Explain why function notation notation. In function notation. T d makes it clearer that T is a function of d. The equation represents a function. I knew that the relationship was linear because the temperature increases at a steady rate. For the new mine. I substituted the d-values into the function equation to get the T d -values.
I wanted the temperature when d 5 I substituted for d in the equation. I used d for depth and called the function T d for temperature. This is a function because it is a linear relationship. I found that T was about By extrapolating. Xscl B-2 and B The temperature at the bottom is T Then I joined them with a straight line.
East Rand mine is m deep. Temperature of a Mine y 90 80 Temperature. Tech Support For help using a graphing calculator to graph and evaluate functions.
Yscl I graphed the function by entering Y1 5 11 1 0. It was approximately I interpolated to read T from the graph. Using a Graphing Calculator a Let T d represent the temperature in degrees Celsius at a depth of d metres. This told me that T 5 I found that T 5 Reflecting A. Then I used the value operation again to find the temperature at the bottom of the other mine.
How did Lucy use the function equation to determine the two temperatures? What does T mean? How did Stuart use the graph to determine the value of T ? As a check. How did Lucy. I called up the function on my calculator home screen. I chose x to be the original number. The person with the highest final number won the pizza slice.
Rhea 22 b The original numbers chosen by the family members are shown. Multiply by the original number x 12 2x output f x 5 x 12 2 2x 5 12x 2 2x 2 The expression for the final answer is quadratic. I chose f x as the name for the final answer. Each person had to think of a number. Who won the pizza slice? Sara 7 c What would be the best number to choose? This meant that this quadratic function had a maximum value at its vertex. I checked my answer by graphing.
I remembered that the x-coordinate of the vertex is halfway between the two x-intercepts. I recognized that the equation was quadratic and that its graph would be a parabola that opened down. I put the equation back in factored form by dividing out the common factor. When x 5 3. I drew a line up to the graph from the x-axis at x 5 3 and then a line across from that point of intersection to the y-axis.
I saw that there was no graph to the left of the point The x-value is 0 at this point. I saw that the graph crosses the y-axis at y 5 1.
The domain is all real numbers greater than or equal to The y-value was 2. The range is all real numbers greater than or equal to 0. I simplified the equation. I used square brackets to keep the functions separate until I had simplified each one.
I substituted c 1 2 for x in both functions. For this reason. To evaluate f a. The output depends on the equation of the function. There is 1. Consider the function f s 5 s 2 2 6s 1 9. What does f 22 represent on the graph of f? State the domain and range of the relation. How do you know that f is a function from its graph?
Consider the function g t 5 3t 1 5. The graph of y 5 f x is shown at the right. Evaluate f ? As a mental arithmetic exercise. Write a function to model the arch. Include a discussion of the advantages of using function notation. Determine the values of x for which Indicate on the graph how you would find f The highest and lowest marks awarded on an examination were and Each arch is set in concrete foundations that are on opposite sides of the St.
All the marks must be reduced so that the highest and lowest marks become and The top of each arch rises 71 m above the river. The arches are m apart. Clair River. A function f x has these properties: The second span of the Bluewater Bridge in Sarnia. Communication Tip family a collection of functions or lines or curves sharing common characteristics parent function the simplest.
This function is called the parent function. Linear Functions Quadratic Functions! The parent functions are in green. In mathematics. Here are some members of the linear and quadratic families. Each member of a family of functions is related to the simplest. How is it different from the graphs of linear and quadratic functions? In your table. Explain how you know that these equations are both functions. Clear all equations from the equation editor.
You can change to these settings by pressing ZOOM 26 4. Is it a function? What are the characteristics of these parent functions that distinguish them from each other?
Make a table like the one shown. NEL Chapter 1. Graph the square root function. Use your graphing calculator to check the sketches shown for f x 5 x and f x 5 x 2 and add anything you think is missing from the descriptions. 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. 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.
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