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

Verified step by step guidance

1. Recall that reflecting a graph across the y-axis means replacing every x-coordinate with its opposite, or -x. This changes the input of the function from x to -x.
2. Start with the original function: \(f(x) = 3x - 4\).
3. To find the reflection across the y-axis, substitute \(-x\) for \(x\) in the function: \(f(-x) = 3(-x) - 4\).
4. Simplify the expression: \(f(-x) = -3x - 4\).
5. The equation \(f(-x) = -3x - 4\) represents the reflection of the graph of \(f(x)\) across the y-axis.

Key Concepts

Function Reflection Across the y-Axis

Reflecting a function across the y-axis involves replacing the input variable x with -x in the function's formula. This transformation flips the graph horizontally, so each point (x, y) moves to (-x, y). For example, if f(x) = 3x - 4, then its reflection is f(-x) = 3(-x) - 4.

Linear Functions and Their Graphs

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is a straight line. Understanding the slope and intercept helps visualize how transformations like reflections affect the line's position and orientation.

Function Notation and Evaluation

Function notation f(x) represents the output value for an input x. Evaluating a function at a specific input means substituting that input into the formula. This concept is essential when applying transformations, such as replacing x with -x to find the reflected function.