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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 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).
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 48 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 8 comma 0 right parenthesis , and the dilated function has an x -intercept of left parenthesis 8 comma 0 right parenthesis .

Answer 1

To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function with a factor \( k = 6 \), we need to analyze both functions.

1. **Original Function \( y = 12x - 4 \)**:
The x-intercept occurs when \( y = 0 \).
\[
0 = 12x - 4 \\
12x = 4 \\
x = \frac{4}{12} = \frac{1}{3}
\]
So, the x-intercept for the original function is \( \left(\frac{1}{3}, 0\right) \).

2. **Vertically Dilated Function**:
A vertical dilation by a factor of \( k \) means multiplying the entire function by \( k \). Thus, the dilated function becomes:
\[
y = 6(12x - 4) = 72x - 24
\]
We find the x-intercept for the dilated function:
\[
0 = 72x - 24 \\
72x = 24 \\
x = \frac{24}{72} = \frac{1}{3}
\]
So, the x-intercept for the dilated function is also \( \left(\frac{1}{3}, 0\right) \).

### Summary of x-intercepts:
- Original function: \( \left(\frac{1}{3}, 0\right) \)
- Dilated function: \( \left(\frac{1}{3}, 0\right) \)

**Conclusion**: 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 \( \left(\frac{1}{3}, 0\right) \)."

However, none of the provided responses match this conclusion, indicating an error in the options given.