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 (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)
.
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 find the x-intercepts of the original function and the vertically dilated function, we start with the original function:

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

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

Now, we solve for \( x \):
\[ \frac{1}{2}x = 4 \]
\[ x = 4 \cdot 2 = 8 \]

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

### Vertically Dilated Function:
The vertically dilated function with a factor of \( k = 6 \) is given by multiplying the original function by 6:
\[ y = 6\left(\frac{1}{2}x - 4\right) \]
\[ y = 3x - 24 \]

Now, we find the x-intercept of the dilated function by setting \( y = 0 \):
\[ 0 = 3x - 24 \]

Solving for \( x \):
\[ 3x = 24 \]
\[ x = \frac{24}{3} = 8 \]

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

### Conclusion:
The x-intercepts are:
- The original function: \( (8, 0) \)
- The dilated function: \( (8, 0) \)

So, 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).**