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 (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 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 (−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 (48,0).

Answer 1

Let's start by finding the x-intercepts of the original function and the vertically dilated function.

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

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

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

Next, let's find the vertically dilated function. The vertically dilated function with a factor \( k = 6 \) is:
\[ y = 6\left(\frac{1}{2}x - 4\right) \]
Let's simplify this:
\[ y = 6 \cdot \frac{1}{2}x - 6 \cdot 4 \]
\[ y = 3x - 24 \]

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

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

In summary:
- The original function has an x-intercept of \( (8, 0) \).
- The dilated function 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).**