Question

Use a graphing calculator to graph the original function y=12x−4 and the vertically dilated function with a factor k=6. Determine the x-intercepts of both functions.(1 point)
Responses

The original function has an x-intercept of (0,−4), and the dilated function has an x-intercept of (0,−24).
The original function has an x -intercept of left parenthesis 0 comma negative 4 right parenthesis , and the dilated function has an x -intercept of left parenthesis 0 comma negative 24 right parenthesis .

The original function has an x-intercept of (8,0), and the dilated function has an x-intercept of (8,0).
The original function has an x -intercept of left parenthesis 8 comma 0 right parenthesis , and the dilated function has an x -intercept of left parenthesis 8 comma 0 right parenthesis .

The original function has an x-intercept of (−8,0), and the dilated function has an x-intercept of (−8,0).
The original function has an x -intercept of left parenthesis negative 8 comma 0 right parenthesis , and the dilated function has an x -intercept of left parenthesis negative 8 comma 0 right parenthesis .

The original function has an x-intercept of (8,0), and the dilated function has an x-intercept of (48,0).

Answer 1

To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function \( y = k(12x - 4) \) with a factor \( k = 6 \), we can proceed as follows:

1. **Find the x-intercept of the original function \( y = 12x - 4 \)**:
- Set \( y = 0 \):
\[
0 = 12x - 4
\]
- Solve for \( x \):
\[
12x = 4 \quad \Rightarrow \quad x = \frac{4}{12} = \frac{1}{3}
\]
- The x-intercept of the original function is \( \left( \frac{1}{3}, 0 \right) \).

2. **Find the x-intercept of the vertically dilated function \( y = 6(12x - 4) \)**:
- Set \( y = 0 \):
\[
0 = 6(12x - 4)
\]
- Since \( 6 \) is a multiplicative factor, we only need to solve the equation inside the parentheses:
\[
0 = 12x - 4
\]
- This is the same equation that we solved earlier:
\[
12x = 4 \quad \Rightarrow \quad x = \frac{4}{12} = \frac{1}{3}
\]
- The x-intercept of the dilated function is also \( \left( \frac{1}{3}, 0 \right) \).

Both functions have the same x-intercept.

Thus, the correct response is:
**The original function has an x-intercept of \( \left( \frac{1}{3}, 0 \right) \), and the dilated function has an x-intercept of \( \left( \frac{1}{3}, 0 \right) \).**

(Note: None of the provided options matches the correct answer. The correct intercept for both functions is \( \left( \frac{1}{3}, 0 \right) \).)