Question

Write a function \(g\) whose graph represents the indicated transformation of the graph of \(f\). Use a graphing calculator to check your answer. \(f(x)=2 x+6 ;\) vertical shrink by a factor of \(\frac{1}{2}\)

Short answer

The function \(g(x)\) that represents the indicated transformation of the original function \(f(x) = 2x+6\) by a vertical shrink of \(\frac{1}{2}\) is \(g(x) = x+3\).

Step-by-step solution

Define the base function

The original function is given as \(f(x)=2 x+6\). This is a linear function with slope 2 and y-intercept 6.

Identify the transformation

We're asked to perform a vertical shrink on the graph of the function by a factor of \(\frac{1}{2}\). This means we'll be multiplying the y-values (outputs of the function) by \(\frac{1}{2}\).

Apply the transformation to the function

To apply the vertical shrink, we'll multiply the original function by \(\frac{1}{2}\). So our new function \(g(x)\) will be \(\frac{1}{2}\) times the original function \(f(x)\). Therefore, \(g(x)= \frac{1}{2}f(x)= \frac{1}{2}(2x+6) = x+3\).

Check your work with a graphing calculator

We can use a graphing calculator to ensure our answer is correct. Graph both the original function \(f(x) = 2x + 6\) and the transformed function \(g(x) = x+3\). The resulting graph for \(g(x)\) should look like a vertically shrunken version of the graph for \(f(x)\).

Key concepts

Linear Functions

Linear functions are one of the most fundamental types of functions in mathematics and are represented by the equation form: \(f(x) = mx + b\). The graph of a linear function is a straight line, where:

• \(m\) is the slope of the line, indicating its steepness and the direction it tilts.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In the exercise given, the initial function is \(f(x) = 2x + 6\). Here, the slope \(m\) is 2, and the y-intercept \(b\) is 6.
This means for every unit increase in \(x\), the value of \(f(x)\) increases by 2. The line crosses the y-axis at the point (0, 6).
Understanding these components of a linear function is essential for analyzing the effects of transformations like those discussed in the exercise.

Vertical Stretch and Shrink

Vertical stretch and shrink transformations alter the vertical scale of a function's graph without changing its horizontal properties. To perform a vertical shrink by a factor, each y-value of the function is multiplied by a constant factor \(k\).

• If \(0 < k < 1\), as in the vertical shrink, the graph becomes compressed in the vertical direction.
• If \(k > 1\), the graph stretches vertically.

In the exercise, the transformation involves a vertical shrink by a factor of \(\frac{1}{2}\).
This means all y-values derived from the original function \(f(x) = 2x + 6\) are multiplied by \(\frac{1}{2}\).
The new function, \(g(x) = x + 3\), represents this vertical shrink.
The slope and y-intercept change slightly due to the multiplicative factor, but the general linear nature of the graph is preserved.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing and verifying mathematical transformations.
Through graphing technology, students can see real-time changes in functions and their corresponding graphs, aiding in deeper comprehension.
To use a graphing calculator for this exercise:

• First, input the original function \(f(x) = 2x + 6\) and observe the line passing through (0, 6).
• Next, input the transformed function \(g(x) = x + 3\) to see the effects of the vertical shrink.
• Compare the two graphs to verify that \(g(x)\) is indeed a vertically shrunken version of \(f(x)\).

The graphing calculator effectively confirms that every point on \(g(x)\) is half the height of the point on \(f(x)\) for the same \(x\), ensuring the calculation is correct.

Slope and Y-Intercept

The slope and y-intercept are critical concepts when understanding linear functions and transformations.
The slope \(m\) tells us how steep the line is, while the y-intercept \(b\) shows where it crosses the y-axis.

• A greater slope means a steeper line, while a smaller slope, closer to zero, means a flatter line.
• The y-intercept gives a quick reference for graph placement on the coordinate plane.

In our initial function \(f(x) = 2x + 6\), the slope is 2, providing a moderately steep line, crossing the y-axis at 6.
After the vertical shrink, \(g(x) = x + 3\) gives us a new slope of 1 and a y-intercept of 3, demonstrating how such transformations affect these parameters.
Grasping the influence on slope and y-intercept helps in predicting graph changes following any transformation.