Question

Suppose that the function \(y=f(x)\) is decreasing on the interval [-2,7] (a) Over what interval is the graph of \(y=f(x+2)\) decreasing? (b) Over what interval is the graph of \(y=f(x-5)\) decreasing? (c) Is the graph of \(y=-f(x)\) increasing, decreasing, or neither on the interval [-2,7]\(?\) (d) Is the graph of \(y=f(-x)\) increasing, decreasing, or neither on the interval [-7,2]\(?\)

Short answer

a) Decreasing on \([-4, 5]\). b) Decreasing on \([3, 12]\). c) Increasing on \([-2, 7]\). d) Decreasing on \([-7, 2]\).

Step-by-step solution

Understand the given function and interval

We are given that the function \(y = f(x)\) is decreasing on the interval \([-2, 7]\). This means that for any two points in this interval, if \(x_1 < x_2\), then \(f(x_1) > f(x_2)\).

Part (a): Determine the new interval for \(y = f(x+2)\)

To find where \(y = f(x+2)\) is decreasing, notice that shifting the function by 2 units to the left will shift the interval of decreasing. If \(y = f(x)\) is decreasing on \([-2, 7]\), then \(y = f(x+2)\) is decreasing where \(x+2\) is in \([-2, 7]\). This translates to \([-4, 5]\).

Part (b): Determine the new interval for \(y = f(x-5)\)

Shifting the function by 5 units to the right means altering the interval of decreasing in the same way. If \(y = f(x)\) is decreasing on \([-2, 7]\), then \(y = f(x-5)\) decreases on the interval where \(x-5\) is in \([-2, 7]\). This translates to \([3, 12]\).

Part (c): Determine if the graph of \(y = -f(x)\) is increasing or decreasing

Reflecting the function around the x-axis changes the direction of increase and decrease. If \(y = f(x)\) is decreasing on \([-2, 7]\), then \(y = -f(x)\) is increasing on the same interval \([-2, 7]\).

Part (d): Determine if the graph of \(y = f(-x)\) is increasing or decreasing

Reflecting the function around the y-axis needs special attention. If \(y = f(x)\) is decreasing on \([-2, 7]\), then \(y = f(-x)\) will be decreasing on the interval \([-7, 2]\) because reflecting doesn't change the decreasement property, just the interval over which it applies.

Key concepts

interval notation

Interval notation is a way of describing subsets of the real number line. It uses brackets and parentheses to indicate whether endpoints are included or excluded. For example, the notation \([-2, 7]\) includes all values of x from -2 to 7, inclusive.
There are two types of intervals:

• Closed interval: [a, b] includes both endpoints a and b.
• Open interval: (a, b) excludes both endpoints a and b.

In problems involving decreasing functions, it's crucial to identify the correct interval where the function maintains its decreasing behavior.

function transformations

Function transformations involve moving, scaling, stretching, or reflecting the graph of a function. There are several types of transformations:

• Translation: Shifts the graph horizontally or vertically.
• Scaling: Expands or compresses the graph vertically or horizontally.
• Reflection: Flips the graph over a given axis.
• Rotation: Rotates the graph around a point.

In this exercise, the transformations include horizontal shifting, as seen in parts (a) and (b). For instance, shifting a function such as y = f(x) to the left by 2 units results in y = f(x+2), while shifting it to the right by 5 units results in y = f(x-5). These shifts alter the intervals over which the function is increasing or decreasing.

reflection of functions

Reflecting a function involves flipping it over a specified axis.

• Reflection over the x-axis: Changes y = f(x) to y = -f(x). This flips the graph upside down, such that increasing sections become decreasing and vice versa.
• Reflection over the y-axis: Changes y = f(x) to y = f(-x). This flips the graph over the y-axis, reversing the direction but keeping the increasing and decreasing properties the same on the reflected interval.

For example, in part (c) of the exercise, reflecting y = f(x) over the x-axis results in y = -f(x), changing a decreasing function to an increasing one on the interval [-2, 7]. Similarly, part (d) reflects y = f(x) over the y-axis to become y = f(-x), maintaining the decreasing nature over a different interval, from [-7, 2].

shifting graphs

Shifting the graph of a function means moving it horizontally or vertically without altering its shape. In this exercise, we mostly deal with horizontal shifts:

• For y = f(x+2), the graph of y = f(x) is shifted 2 units to the left, altering the interval of decreasing behavior.
• For y = f(x-5), the graph of y = f(x) is shifted 5 units to the right, changing the interval once again.

To determine the new intervals of decreasing behavior:
- For y = f(x+2) decreasing, solve for x+2 within the original interval \([-2, 7]\). This gives us \([-4, 5]\).
- For y = f(x-5) decreasing, solve for x-5 within \([-2, 7]\). This translates to \([3, 12]\).
Understanding how shifting impacts the intervals helps us better grasp the behavior of the transformed functions.