Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Transformations

College Algebra

3. Functions

Transformations

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1:20 minutes

Problem 92

Textbook Question

What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

Verified step by step guidance

Understand that the function \( f(x) = |x| \) represents the absolute value of \( x \), which is a V-shaped graph opening upwards with its vertex at the origin (0,0).

Recognize that the function \( g(x) = |-x| \) is also an absolute value function, but it takes the absolute value of \( -x \).

Realize that \( |-x| = |x| \) because the absolute value function removes any negative sign, making \( g(x) \) identical to \( f(x) \).

Conclude that the graphs of \( f(x) = |x| \) and \( g(x) = |-x| \) are the same, both being V-shaped and symmetric about the y-axis.

Visualize that there is no transformation such as translation, reflection, or scaling between the graphs of \( f(x) \) and \( g(x) \); they overlap completely.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means that for any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. The graph of ƒ(x) = |x| is a V-shaped curve that opens upwards, with its vertex at the origin (0,0).

Reflection Across the Y-Axis

The function g(x) = |-x| represents a reflection of the function ƒ(x) = |x| across the y-axis. This is because replacing x with -x in the function does not change the output of the absolute value, as both positive and negative inputs yield the same result. Thus, the graphs of ƒ(x) and g(x) are identical, demonstrating symmetry about the y-axis.

Graphical Symmetry

Graphical symmetry refers to the property of a graph where one side mirrors the other across a specific line, such as the y-axis. In this case, both ƒ(x) = |x| and g(x) = |-x| exhibit this symmetry, as their graphs are the same. Understanding this concept helps in analyzing how transformations affect the shape and position of graphs in the coordinate plane.