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 (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 .

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)
.

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 start by finding the x-intercept of the original function.

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

2. **Finding the equation of the vertically dilated function with a factor \(k = 6\)**:
The vertically dilated function can be represented as:
\[
y = k(12x - 4) = 6(12x - 4) = 72x - 24
\]

3. **Finding x-intercept of the dilated function \(y = 72x - 24\)**:
Again, setting \(y = 0\):
\[
0 = 72x - 24
\]
Solving for \(x\):
\[
72x = 24 \quad \Rightarrow \quad x = \frac{24}{72} = \frac{1}{3}
\]
So the x-intercept of the dilated function is also \(\left(\frac{1}{3}, 0\right)\).

Based on these calculations, the responses regarding the x-intercepts are as follows:

- The original function has an x-intercept of \(\left(\frac{1}{3}, 0\right)\)
- The dilated function also has an x-intercept of \(\left(\frac{1}{3}, 0\right)\)

None of the options listed in your original question accurately represent the correct x-intercepts, as they either refer to different x-intercepts or incorrect values. Both functions share the same x-intercept \(\left(\frac{1}{3}, 0\right)\).