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 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 (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)
.
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 (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 .

Answer 1

To find the x-intercepts of the original function and the vertically dilated function, let’s first analyze them one by one.

### Original Function:
The original function is given as:
\[ y = \frac{1}{2}x - 4 \]

To find the x-intercept, we set \( y = 0 \):
\[ 0 = \frac{1}{2}x - 4 \]
\[ \frac{1}{2}x = 4 \]
\[ x = 4 \times 2 \]
\[ x = 8 \]

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

### Vertically Dilated Function:
The vertically dilated function with a factor \( k = 6 \) can be expressed as:
\[ y = 6 \left( \frac{1}{2}x - 4 \right) \]
This simplifies to:
\[ y = 3x - 24 \]

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

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

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

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).**