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 (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 (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)

Answer 1

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

1. **Original Function:**
Set \( y = 0 \) to find the x-intercept:
\[
0 = 12x - 4
\]
Solving for \( x \):
\[
12x = 4 \\
x = \frac{4}{12} = \frac{1}{3}
\]
Therefore, the x-intercept of the original function is \( \left(\frac{1}{3}, 0\right) \).

2. **Vertically Dilated Function:**
The vertically dilated function with a factor of \( k = 6 \) can be written as:
\[
y = 6(12x - 4) = 72x - 24
\]
To find the x-intercept, set \( y = 0 \):
\[
0 = 72x - 24
\]
Solving for \( x \):
\[
72x = 24 \\
x = \frac{24}{72} = \frac{1}{3}
\]
Thus, the x-intercept of the dilated function is also \( \left(\frac{1}{3}, 0\right) \).

### Conclusion:
Both the original function \( y = 12x - 4 \) and the dilated function \( y = 72x - 24 \) have the same x-intercept at \( \left(\frac{1}{3}, 0\right) \).

The options you provided do not seem to reflect this outcome. However, the accurate conclusion based on calculations is that both functions have an x-intercept of \( \left(\frac{1}{3}, 0\right) \).