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 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 .

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

Answer 1

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

### 1. Finding the x-intercept of the original function
The x-intercept occurs where \( y = 0 \). For the original function:
\[
0 = 12x - 4
\]
Adding 4 to both sides:
\[
4 = 12x
\]
Dividing both sides by 12:
\[
x = \frac{4}{12} = \frac{1}{3}
\]

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

### 2. Finding the x-intercept of the vertically dilated function
The vertically dilated function with a factor of \( k = 6 \) is:
\[
y = 6(12x - 4) = 72x - 24
\]
To find the x-intercept for this function:
\[
0 = 72x - 24
\]
Adding 24 to both sides:
\[
24 = 72x
\]
Dividing both sides by 72:
\[
x = \frac{24}{72} = \frac{1}{3}
\]

Thus, the x-intercept of the dilated function is:
\[
\left(\frac{1}{3}, 0\right)
\]

### Conclusion
After calculating, we find that:
- The original function \( y = 12x - 4 \) has an x-intercept of \(\left(\frac{1}{3}, 0\right)\)
- The vertically dilated function \( y = 72x - 24 \) also has an x-intercept of \(\left(\frac{1}{3}, 0\right)\)

None of the responses provided match the correct x-intercept calculations.