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)
.
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 .

Answer 1

To find the x-intercepts of both the original function and the vertically dilated function, we will first rewrite the original function and then find the x-intercepts.

The original function is:
\[ y = \frac{1}{2}x - 4. \]

To find the x-intercept, set \( y \) to 0 and solve for \( x \):
\[ 0 = \frac{1}{2}x - 4. \]
\[ \frac{1}{2}x = 4. \]
\[ x = 4 \cdot 2 = 8. \]

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

Next, we consider the vertically dilated function with a factor of \( k = 6 \). The vertically dilated version of the function is given by multiplying the function by \( k \):
\[ y = 6 \left( \frac{1}{2}x - 4 \right) = 3x - 24. \]

Now we find the x-intercept of the dilated function by setting \( y \) to 0:
\[ 0 = 3x - 24. \]
\[ 3x = 24. \]
\[ x = \frac{24}{3} = 8. \]

Thus, the x-intercept of the dilated function is:
\[ (8, 0). \]

### Summary of Results:
- The original function has an x-intercept of \( (8, 0) \).
- The dilated function has an x-intercept of \( (8, 0) \).

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