Question

Use a graphing calculator to graph the original function: y = 1/2 x - 4, and the vertically dilated function with a factor: k = 6. Determine the x-intercepts of both functions.
A. The original function has an x-intercept of: (8, 0), and the dilated function has an x-intercept of: (48, 0).
B. The original function has an x-intercept of: (8, 0), and the dilated function has an x-intercept of: (8, 0).
C. The original function has an x-intercept of, (-8, 0), and the dilated function has an x-intercept of: (-8, 0)
D. The original function has an x-intercept of, (0, -4), and the dilated function has an x-intercept of: (0, -24)

Answer 1

To answer your question, let's first find the x-intercept of the original function \( y = \frac{1}{2} x - 4 \).

The x-intercept occurs when \( y = 0 \):
\[
0 = \frac{1}{2} x - 4
\]
To solve for \( x \):
\[
\frac{1}{2} x = 4
\]
\[
x = 8
\]
So, the x-intercept of the original function is \( (8, 0) \).

Next, let's determine the dilated function. A vertical dilation by a factor of \( k = 6 \) means we multiply the entire function by 6. The vertically dilated function is:
\[
y = 6\left(\frac{1}{2} x - 4\right) = 3x - 24
\]

Now, we find the x-intercept of the dilated function \( y = 3x - 24 \):
\[
0 = 3x - 24
\]
To solve for \( x \):
\[
3x = 24
\]
\[
x = 8
\]
So, the x-intercept of the dilated function is also \( (8, 0) \).

Now, we can summarize the findings:
- The original function has an x-intercept of: \( (8, 0) \)
- The dilated function has an x-intercept of: \( (8, 0) \)

Therefore, the correct answer is:
**B. The original function has an x-intercept of: (8, 0), and the dilated function has an x-intercept of: (8, 0).**