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

Answer 1

To graph the original function \( y = \frac{1}{2}x - 4 \) and the vertically dilated function with a factor of \( k = 6 \), we first need to write the equation for the vertically dilated function.

### Step 1: Write the vertically dilated function
A vertical dilation by a factor of \( k \) means that we multiply the entire function by \( k \). Therefore, the vertically dilated function is:

\[
y = 6 \left( \frac{1}{2}x - 4 \right) = 3x - 24
\]

### Step 2: Find the x-intercepts
The x-intercept occurs when \( y = 0 \).

#### For the original function \( y = \frac{1}{2}x - 4 \):
Set \( y = 0 \):

\[
0 = \frac{1}{2}x - 4
\]

Add 4 to both sides:

\[
4 = \frac{1}{2}x
\]

Multiply both sides by 2:

\[
x = 8
\]

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

#### For the vertically dilated function \( y = 3x - 24 \):
Set \( y = 0 \):

\[
0 = 3x - 24
\]

Add 24 to both sides:

\[
24 = 3x
\]

Divide both sides by 3:

\[
x = 8
\]

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

### Summary:
- The x-intercept of the original function \( y = \frac{1}{2}x - 4 \) is \( (8, 0) \).
- The x-intercept of the vertically dilated function \( y = 3x - 24 \) is also \( (8, 0) \).

Both functions have the same x-intercept at \( x = 8 \).