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 (−8,0), and the dilated function has an x-intercept of (−8,0)

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 (8,0), and the dilated function has an x-intercept of (8,0)

Answer 1

To find the x-intercepts of the functions \( y = 12x - 4 \) and its vertically dilated version, we first need to determine the x-intercept of the original function.

### Finding the x-intercept of the original function:

Set \( y = 0 \):

\[
0 = 12x - 4
\]

Add 4 to both sides:

\[
4 = 12x
\]

Now, divide both sides by 12:

\[
x = \frac{4}{12} = \frac{1}{3}
\]

Thus, the x-intercept of the original function is \( \left( \frac{1}{3}, 0 \right) \).

### Finding the vertically dilated function:

The vertically dilated function with a factor \( k = 6 \) is given by multiplying the original function by \( 6 \):

\[
y = 6(12x - 4) = 72x - 24
\]

### Finding the x-intercept of the dilated function:

Set \( y = 0 \):

\[
0 = 72x - 24
\]

Add 24 to both sides:

\[
24 = 72x
\]

Now, divide both sides by 72:

\[
x = \frac{24}{72} = \frac{1}{3}
\]

### Conclusion on the x-intercepts:

The x-intercepts are:
- For the original function: \( \left( \frac{1}{3}, 0 \right) \)
- For the dilated function: \( \left( \frac{1}{3}, 0 \right) \)

### Response:

The correct option is:

**The original function has an x-intercept of \( ( \frac{1}{3}, 0 ) \), and the dilated function has an x-intercept of \( ( \frac{1}{3}, 0 ) \)**.