Chapter 2.5, Problem 37AYU

Suppose that the function $y\text{ = }f\left(x\right)$ is increasing on the interval $\left[-1,\text{ 5}\right]$.

(a) Over what interval is the graph of $y\text{ = }f\left(x\text{ + 2}\right)$ increasing?

(b) Over what interval is the graph of $y\text{ = }f\left(x\text{ - 5}\right)$ increasing?

(c) What can be said about the graph of $y\text{ = -}f\left(x\right)$?

(d) What can be said about the graph of $y\text{ = }f\left(-x\right)$?

To determine

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

a. Over what interval is the graph of $y=f\left(x+2\right)$ increasing?

a. The function $y=f\left(x+2\right)$ is increasing on the interval $\left[-3,3\right]$.

Explanation:

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

a. The graph of $y=f\left(x+2\right)$ increasing.

Horizontally shift left 2 units; subtract 2 from both $\left[-1,5\right]$.

We get $\left[-3,3\right]$.

The function $y=f\left(x+2\right)$ is increasing on the interval $\left[-3,3\right]$.

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

b. Over what interval is the graph of $y=f\left(x-5\right)$ increasing?

b. The function $y=f\left(x-5\right)$ is increasing on the interval $\left[4,10\right]$.

Explanation:

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

b. The graph of $y=f\left(x-5\right)$ increasing.

Horizontally shift right 5 units; add 5 to both $\left[-1,5\right]$.

We get $\left[4,10\right]$.

The function $y=f\left(x-5\right)$ is increasing on the interval $\left[4,10\right]$.

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

c. What can be said about the graph of $y=-f\left(x\right)$?

c. The function $y=-f\left(x\right)$ is decreasing on the interval $\left[-1,5\right]$.

Explanation:

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

c. The graph of $y=-f\left(x\right)$.

Reflection about the $x\text{-axis}$.

We get $\left[-1,5\right]$.

The function $y=-f\left(x\right)$ is decreasing on the interval $\left[-1,5\right]$.

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

d. What can be said about the graph of $y=f\left(-x\right)$?

d. The function $y=f\left(-x\right)$ is decreasing on the interval $\left[-5,1\right]$.

Explanation:

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

d. The graph of $y=f\left(-x\right)$.

Reflection about the $y\text{-axis}$.

We get $\left[-5,1\right]$.

The function $y=f\left(-x\right)$ is decreasing on the interval $\left[-5,1\right]$.

Answer to Problem 37AYU

a. The function $y=f\left(x+2\right)$ is increasing on the interval $\left[-3,3\right]$.

Explanation of Solution

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

a. The graph of $y=f\left(x+2\right)$ increasing.

Horizontally shift left 2 units; subtract 2 from both $\left[-1,5\right]$.

We get $\left[-3,3\right]$.

The function $y=f\left(x+2\right)$ is increasing on the interval $\left[-3,3\right]$.

To determine

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

b. Over what interval is the graph of $y=f\left(x-5\right)$ increasing?

Answer to Problem 37AYU

b. The function $y=f\left(x-5\right)$ is increasing on the interval $\left[4,10\right]$.

Explanation of Solution

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

b. The graph of $y=f\left(x-5\right)$ increasing.

Horizontally shift right 5 units; add 5 to both $\left[-1,5\right]$.

We get $\left[4,10\right]$.

The function $y=f\left(x-5\right)$ is increasing on the interval $\left[4,10\right]$.

To determine

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

c. What can be said about the graph of $y=-f\left(x\right)$?

Answer to Problem 37AYU

c. The function $y=-f\left(x\right)$ is decreasing on the interval $\left[-1,5\right]$.

Explanation of Solution

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

c. The graph of $y=-f\left(x\right)$.

Reflection about the $x\text{-axis}$.

We get $\left[-1,5\right]$.

The function $y=-f\left(x\right)$ is decreasing on the interval $\left[-1,5\right]$.

To determine

To find: Suppose that the function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

d. What can be said about the graph of $y=f\left(-x\right)$?

Answer to Problem 37AYU

d. The function $y=f\left(-x\right)$ is decreasing on the interval $\left[-5,1\right]$.

Explanation of Solution

Given:

The function $y=f\left(x\right)$ is increasing on the interval $\left[-1,5\right]$.

Calculation:

d. The graph of $y=f\left(-x\right)$.

Reflection about the $y\text{-axis}$.

We get $\left[-5,1\right]$.

The function $y=f\left(-x\right)$ is decreasing on the interval $\left[-5,1\right]$.