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Chapter 6: Problem 60

Graph both functions in the same viewing window and describe how \(g\) is a transformation of $f . $$f(x)=x^{3}, g(x)=-x^{3}$$

Short Answer

Expert verified
The function \(g(x)=-x^3\) is a reflection of \(f(x)=x^3\) across the x-axis.

Step by step solution

01

Understand the Functions

Identify the given functions: the original function is \( f(x) = x^3 \) and the transformed function is \( g(x) = -x^3 \).
02

Analyze the Parent Function

Graph the parent function \( f(x) = x^3 \). This is a cubic function that passes through the origin (0,0). The graph is symmetric about the origin and it increases from left to right.
03

Analyze the Transformation

Note that \( g(x) = -x^3 \) introduces a reflection. Specifically, it reflects the parent function \( f(x) = x^3 \) across the x-axis.
04

Graph the Transformed Function

Graph \( g(x) = -x^3 \) in the same window as \( f(x) = x^3 \). This graph will also pass through the origin (0,0) but will reflect downward through other points compared to \( f(x) = x^3 \).
05

Compare the Functions

Describe how the transformation occurred: \( g(x) \) is the reflection of \( f(x) \) over the x-axis. Every value of \( f(x) \) is multiplied by -1 to form \( g(x) \). Since \( f(x) \) produces positive output values for positive input values, \( g(x) \) produces negative output values for the same input values.

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Key Concepts

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

Cubic Function
A cubic function is a type of polynomial function of degree three. Its general form is given by \( f(x) = ax^3 + bx^2 + cx + d \). In the given exercise, the function is \( f(x) = x^3 \), which is a simple cubic function where the coefficients b, c, and d are all zero. This results in a graph that passes through the origin (0,0).
The graph of a cubic function has a few key characteristics:
  • It is symmetric about the origin.
  • It can pass through the origin as it does in the case of our specific function \( f(x) = x^3 \).
  • As \( x \) approaches positive or negative infinity, the output values of \( f(x) \) also approach positive or negative infinity, respectively.
Understanding these properties helps in visualizing and graphing cubic functions effectively.
Reflection over the x-axis
Reflection over the x-axis is a type of transformation applied to functions. When we reflect a function over the x-axis, we multiply its output by -1. Therefore, if \( f(x) \) is the original function, its reflection over the x-axis is given by \( g(x) = -f(x) \).
In our exercise, the original function is \( f(x) = x^3 \), and its reflected function is \( g(x) = -x^3 \).
This reflection means that for every \( x \)-value in the domain:
  • If \( f(x) \) produces a positive output, then \( g(x) \) will produce a corresponding negative output.
  • Conversely, if \( f(x) \) produces a negative output, then \( g(x) \) will produce a positive output.
This transformation completely flips the graph of \( f(x) \) across the x-axis, making it a mirror image of the original function.
Graphing Functions
Graphing functions involves plotting the points that satisfy the function's equation on a coordinate plane.
When graphing \( f(x) = x^3 \), we observe certain key points:
  • At \( x = 0 \), \( f(x) = 0^3 = 0 \) (origin).
  • At \( x = 1 \, f(x) = 1^3 = 1 \).
  • At \( x = -1, f(x) = (-1)^3 = -1 \).
Given this, we can identify the overall trend of the function.
When graphing \( g(x) = -x^3 \), we plot similar points but apply the reflection transformation:
  • At \( x = 0 \), \( g(x) = -(0^3) = 0 \) (origin).
  • At \( x = 1, g(x) = -(1^3) = -1 \).
  • At \( x = -1, g(x) = -(-1^3) = 1 \).
This gives us the mirrored image of \( f(x) = x^3 \).
Graphing both functions on the same coordinate plane allows us to clearly see the reflection transformation across the x-axis.

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Most popular questions from this chapter

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=-\frac{3}{2} \csc x$$

Find the signs of the six trigonometric function values for the given angles. $$-620^{\circ}$$

Find the signs of the six trigonometric function values for the given angles. $$-57^{\circ}$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$y=x^{2}-x^{3}[4.1]$$

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=\cot (2 x)$$
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