Question

The original function y = 1/2x - 4 and the vertically dilated function with a factor k = 6. Determine the x intercepts of both functions
The original function has and intercept of (8,0) and the dilated function has an x intercept of (8,0)
x intercept of (8,0) and dilated function has an x intercept of (48,0)
x intercept of (-8,0) and dilated function has an x intercept of (-8,0)
x intercept (0,-4) and dilated function has an x intercept of (0,-24)

Answer 1

To determine the x-intercepts of both the original function \( y = \frac{1}{2}x - 4 \) and the vertically dilated function with a factor of \( k = 6 \), we can start by finding the x-intercept of the original function.

### Original Function:
To find the x-intercept, we set \( y = 0 \):

\[
0 = \frac{1}{2}x - 4
\]

Now, we solve for \( x \):

\[
\frac{1}{2}x = 4
\]

Multiplying both sides by 2:

\[
x = 8
\]

Thus, the x-intercept of the original function \( y = \frac{1}{2}x - 4 \) is \( (8, 0) \).

### Vertically Dilated Function:
The vertically dilated function can be represented as:

\[
y = k \left( \frac{1}{2}x - 4 \right) = 6 \left( \frac{1}{2}x - 4 \right)
\]

Distributing the \( k = 6 \):

\[
y = 3x - 24
\]

Now, we find the x-intercept of this dilated function by setting \( y = 0 \):

\[
0 = 3x - 24
\]

Now, we solve for \( x \):

\[
3x = 24
\]

Dividing both sides by 3:

\[
x = 8
\]

Thus, the x-intercept of the dilated function \( y = 3x - 24 \) is also \( (8, 0) \).

### Summary of X-Intercepts:
- Original function \( y = \frac{1}{2}x - 4 \): x-intercept \( (8, 0) \)
- Vertically dilated function \( y = 3x - 24 \): x-intercept \( (8, 0) \)

### Conclusion:
Both functions have the same x-intercept, which is \( (8, 0) \). The correct answer from the options provided is:

- **The original function has an x-intercept of (8,0) and the dilated function has an x-intercept of (8,0)**.