Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x). across the x-axis

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Recall that reflecting a function across the x-axis changes the sign of the output values. This means if the original function is \( f(x) \), the reflected function will be \( -f(x) \).

Start with the given function: \( f(x) = 3x - 4 \).

To find the reflection across the x-axis, multiply the entire function by \( -1 \): \( -f(x) = -(3x - 4) \).

Distribute the negative sign inside the parentheses: \( -f(x) = -3x + 4 \).

Write the equation of the reflected function as \( g(x) = -3x + 4 \), which represents the reflection of \( f(x) \) across the x-axis.

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Key Concepts

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

Function Reflection Across the x-axis

Reflecting a function across the x-axis involves changing the sign of the output values. If the original function is f(x), its reflection is given by -f(x), which flips the graph vertically over the x-axis.

Linear Functions

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. Understanding the structure of linear functions helps in manipulating and graphing them, including transformations like reflections.

Graph Transformations

Graph transformations modify the graph of a function without changing its basic shape. Reflections, translations, stretches, and compressions are common transformations that help visualize how the function changes under different operations.

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Related Practice

Textbook Question

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Textbook Question

Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x). across the y-axis

Textbook Question

For each function graphed, give the minimum and maximum values of ƒ(x) and the x-values at which they occur.

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Textbook Question

For each function graphed, give the minimum and maximum values of ƒ(x) and the x-values at which they occur.

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Textbook Question