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Chapter 3: Problem 75

Suppose that the function \(y=f(x)\) is increasing on the interval [-1,5] (a) Over what interval is the graph of \(y=f(x+2)\) increasing? (b) Over what interval is the graph of \(y=f(x-5)\) increasing? (c) Is the graph of \(y=-f(x)\) increasing, decreasing, or neither on the interval [-1,5]\(?\) (d) Is the graph of \(y=f(-x)\) increasing, decreasing, or neither on the interval [-5,1]\(?\)

Short Answer

Expert verified
(a) [-3,3], (b) [4,10], (c) Decreasing, (d) Decreasing

Step by step solution

01

- Analyze the given function for part (a)

Given that the function y=f(x) is increasing on the interval [-1,5], identify the change in the argument for function f in part (a), which is x + 2. This translates every x value in f by -2.
02

- Determine the interval for part (a)

For the function y=f(x+2), the original interval [-1,5] shifts left by 2 units. Therefore, convery each x value in [-1,5] by subtracting 2. [-1-2 , 5-2] = [-3 , 3]
03

- Analyze the given function for part (b)

Given the function y=f(x) is increasing on the interval [-1,5], identify the change in the argument for function (b) which is f(x-5), moves every x value in f by +5.
04

- Determine the interval for part (b)

For y=f(x-5), the original interval [-1,5] shifts right by 5 units. Therefore, convey each x value in [-1,5] by adding 5. [-1+5 , 5+5] = [4 , 10]
05

- Analyze the graph transformation for part (c)

Given the function y=f(x) is increasing, for the function y=-f(x), negate the value of the original increasing function which leads to a decreasing function.
06

- Determine the behavior for part (c)

Since y=f(x) is increasing on [-1,5], the function y=-f(x) will be decreasing on the same interval [-1,5].
07

- Analyze the graph transformation for part (d)

Given the function y=f(x) is increasing, for the function y=f(-x) which rotates the graph across the y-axis.
08

- Determine the behavior for part (d)

Reversing the x values for f(-x), changes the axis on which f is evaluated. The function y=f(x) being increasing as x increases, results the negation, making y=f(-x) decreasing on interval [-5,1] which is the reflected of [-1,5].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Transformations
Function transformations refer to shifting, stretching, compressing, and reflecting the graph of a basic function. These transformations help analyze how changes to the function equation affect its graph.
The exercise explores the transformations involving horizontal shifts and reflections:
  • Shifting the function horizontally affects the x-values. For instance, with an equation like y = f(x + 2), every point on the graph of y = f(x) is shifted left by 2 units.
  • Reflections involve flipping the graph across an axis. For example, y = -f(x) reflects the graph of y = f(x) across the x-axis, while y = f(-x) reflects it across the y-axis.
Understanding these foundational transformations helps in grasping more complex changes and how they influence the graph鈥檚 behavior.
Increasing and Decreasing Functions
An increasing function means that as x-values increase, the y-values also increase. In contrast, a decreasing function indicates that as x-values increase, the y-values decrease.
For example, given that y = f(x) is increasing on the interval [-1, 5], this suggests:
  • As x moves from -1 to 5, the value of f(x) continually increases.
  • In part (c) of the exercise, y = -f(x) becomes decreasing because negating an increasing function flips its values. Therefore, on interval [-1, 5], y = -f(x) decreases.
Understanding these aspects is critical when analyzing function transformations because the behavior of the original function determines the behavior of the transformed one.
Intervals of Functions
The interval of a function is a set range of x-values over which the function is observed or analyzed. Understanding intervals helps determine where a function is increasing or decreasing.
Here鈥檚 how intervals were used in the exercise:
  • In part (a), for y = f(x + 2), each point in interval [-1, 5] shifts left by 2 units, resulting in a new interval [-3, 3].
  • In part (b), for y = f(x - 5), every point in interval [-1, 5] shifts right by 5 units, resulting in the new interval [4, 10].
  • Part (d) involves reflection, thus, the interval [-5, 1] is considered due to the reflection's effect.
Recognizing how these intervals shift or alter with transformations is fundamental for accurate function analysis.
Graph Behavior
Understanding graph behavior involves recognizing specific patterns and attributes of a function's graph, such as increasing, decreasing, and how transformations modify the graph.
In the exercise:
  • Part (a) required analyzing behavior by shifting the graph horizontally to check where it increases.
  • Part (b) involved determining the effect of a horizontal shift on the interval鈥檚 endpoints to see where the function increases.
  • Part (c) considered the behavior change when negating the function, turning an increasing function into a decreasing one.
  • Part (d) analyzed how reflecting the function across the y-axis affects its increasing nature, leading it to decrease where it initially increased.
By understanding graph behavior, you can accurately interpret and predict how transformations affect functions, making this knowledge indispensable in algebra.

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Most popular questions from this chapter

The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) scales for measuring temperature is given by the equation $$ F=\frac{9}{5} C+32 $$ The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and \(\mathrm{Kelvin}(\mathrm{K})\) scales is \(K=C+273 .\) Graph the equation \(F=\frac{9}{5} C+32\) using degrees Fahrenheit on the \(y\) -axis and degrees Celsius on the \(x\) -axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures.

Use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. \(f(x)=-0.4 x^{3}+0.6 x^{2}+3 x-2 \quad[-4,5]\)

Use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. \(f(x)=x^{4}-x^{2} \quad[-2,2]\)

Find the average rate of change of \(g(x)=x^{3}-4 x+7\) : (a) From -3 to -2 (b) From -1 to 1 (c) From 1 to 3

Use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. \(f(x)=-0.4 x^{4}-0.5 x^{3}+0.8 x^{2}-2 \quad[-3,2]\)
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