Identifying a Parent Function In Exercises 2146, is related to one of the parent

Fill in the blanks. Horizontal shifts, vertical shifts, and reflections are called ________ transformations.

Fill in the blanks. A reflection in the -axis of is represented by ________, while a reflection in the -axis of is represented by ________.

Fill in the blanks. A nonrigid transformation of represented by is a ________ ________ when and a ________ ________ when

Match the rigid transformation of with the correct representation of the graph of where (a) (i) A horizontal shift of units to the right (b) (ii) A vertical shift of units down (c) (iii) A horizontal shift of units to the left (d) (iv) A vertical shift of units up

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 1, and 3. (a) (b)

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 1, and 3. (a) (b)

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 0, and 2. (a) (b)

Shifts in the Graph of a Function For each function, sketch (on the same set of coordinate axes) a graph of each function for 1, and 3.

Sketching Transformations In Exercises 9 and 10, use the graph of to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com.

Sketching Transformations In Exercises 9 and 10, use the graph of to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com.

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown. (a) (b)

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown. (a) (b)

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Identifying a Parent Function In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

Identifying a Parent Function In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

Identifying a Parent Function In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

Identifying a Parent Function In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

Identifying a Parent Function In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

Identifying a Parent Function In Exercises 1520, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph.

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

Identifying a Parent Function In Exercises 2146, is related to one of the parent functions described in Section 1.6. (a) Identify the parent function (b) Describe the sequence of transformations from to (c) Sketch the graph of (d) Use function notation to write in terms of f

The shape of , but shifted three units to the right and seven units down

The shape of , but shifted two units to the left, nine units up, and then reflected in the -axis

The shape of , but shifted 13 units to the right

The shape of , but shifted six units to the left, six units down, and then reflected in the -axis

The shape of but shifted 12 units up and then reflected in the -axis

The shape of but shifted four units to the left and eight units down

The shape of but shifted six units to the left and then reflected in both the -axis and the -axis

The shape of but shifted nine units down and then reflected in both the -axis and the -axis

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown. 74 Chapter 1 Functions and Their Graphs

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown. (a) (b)

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown. (a) (b)

Writing Equations from Graphs Use the graph of to write an equation for each function whose graph is shown.

Identifying a Parent Function In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Identifying a Parent Function In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Identifying a Parent Function In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Identifying a Parent Function In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Identifying a Parent Function In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Identifying a Parent Function In Exercises 5964, identify the parent function and the transformation shown in the graph. Write an equation for the function shown in the graph. Then use a graphing utility to verify your answer.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Graphical Analysis In Exercises 6568, use the viewing window shown to write a possible equation for the transformation of the parent function.

Automobile Aerodynamics The number of horsepower required to overcome wind drag on an automobile is approximated by where is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Rewrite the horsepower function so that represents the speed in kilometers per hour. [Find ] Identify the type of transformation applied to the graph of the horsepower function.

Households The numbers (in millions) of households in the United States from 2003 through 2010 can be approximated by where represents the year, with corresponding to 2003. (Source: U.S. Census Bureau) (a) Describe the transformation of the parent function Then use a graphing utility to graph the function over the specified domain. (b) Find the average rate of change of the function from 2003 to 2010. Interpret your answer in the context of the problem. (c) Use the model to predict the number of households in the United States in 2018. Does your answer seem reasonable? Explain.

The graph of is a reflection of the graph of in the -axis.

The graph of is a reflection of the graph of in the -axis.

The graphs of and are identical.

If the graph of the parent function is shifted six units to the right, three units up, and reflected in the -axis, then the point will lie on the graph of the transformation.

Finding Points on a Graph The graph of passes through the points and Find the corresponding points on the graph of

Think About It You can use either of two methods to graph a function: plotting points or translating a parent function as shown in this section. Which method of graphing do you prefer to use for each function? Explain. (a) (b)

Predicting Graphical Relationships Use a graphing utility to graph and in the same viewing window. Before looking at the graphs, try to predict how the graphs of and relate to the graph of f

HOW DO YOU SEE IT? Use the graph of to find the intervals on which each of the graphs in (a)(e) is increasing and decreasing. If not possible, then state the reason. (a) (b) (c) (d) y _ _f _x _ 1_ (e) y _ f _x _ 2_ _ 1

Describing Profits Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function shown. The actual profits are shown by the function along with a verbal description. Use the concepts of transformations of graphs to write in terms of (a) The profits were only three-fourths as large as expected. (b) The profits were consistently $10,000 greater than predicted. (c) There was a two-year delay in the introduction of the product. After sales began, profits grew as expected.

Reversing the Order of Transformations Reverse the order of transformations in Example 2(a). Do you obtain the same graph? Do the same for Example 2(b). Do you obtain the same graph? Explain.