Question

Use a graphing calculator to graph the original function y=1/2x−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 (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 (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 \( y = \frac{1}{2}x - 4 \) and its vertically dilated function \( y = 6(\frac{1}{2}x - 4) \), we first find the x-intercept of the original function.

### Step 1: Find the x-intercept of the original function

An x-intercept occurs when \( y = 0 \).

Setting the original function to zero:
\[
0 = \frac{1}{2}x - 4
\]

Now, solve for \( x \) by adding 4 to both sides:
\[
4 = \frac{1}{2}x
\]

Next, multiply both sides by 2 to isolate \( x \):
\[
x = 8
\]

Thus, the x-intercept of the original function is \( (8, 0) \).

### Step 2: Find the x-intercept of the dilated function

The vertically dilated function with a factor \( k = 6 \) is:
\[
y = 6\left(\frac{1}{2}x - 4\right)
\]

Setting this new function to zero for the x-intercept:
\[
0 = 6\left(\frac{1}{2}x - 4\right)
\]

Since \( 6 \) is not zero, we can simplify the equation:
\[
0 = \frac{1}{2}x - 4
\]

This is the same equation as earlier, which we already solved. Therefore, we get:
\[
4 = \frac{1}{2}x
\]
\[
x = 8
\]

Thus, the x-intercept of the vertically dilated function is also \( (8, 0) \).

### Conclusion

- The original function has an x-intercept of \( (8, 0) \).
- The dilated function also has an x-intercept of \( (8, 0) \).

The correct response to the problem is:
**The original function has an x-intercept of \( (8, 0) \), and the dilated function has an x-intercept of \( (8, 0) \).**