The original function y = 1/2x - 4 and the vertically dilated function with a factor k = 6. Determine the x intercepts of both functions

The original function has and intercept of (8,0) and the dilated function has an x intercept of (8,0)
x intercept of (8,0) and dilated function has an x intercept of (48,0)
x intercept of (-8,0) and dilated function has an x intercept of (-8,0)
x intercept (0,-4) and dilated function has an x intercept of (0,-24)

1 answer

To determine the x-intercepts of both the original function \( y = \frac{1}{2}x - 4 \) and the vertically dilated function with a factor of \( k = 6 \), we can start by finding the x-intercept of the original function.

Original Function:

To find the x-intercept, we set \( y = 0 \):

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

Now, we solve for \( x \):

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

Multiplying both sides by 2:

\[ x = 8 \]

Thus, the x-intercept of the original function \( y = \frac{1}{2}x - 4 \) is \( (8, 0) \).

Vertically Dilated Function:

The vertically dilated function can be represented as:

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

Distributing the \( k = 6 \):

\[ y = 3x - 24 \]

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

\[ 0 = 3x - 24 \]

Now, we solve for \( x \):

\[ 3x = 24 \]

Dividing both sides by 3:

\[ x = 8 \]

Thus, the x-intercept of the dilated function \( y = 3x - 24 \) is also \( (8, 0) \).

Summary of X-Intercepts:

  • Original function \( y = \frac{1}{2}x - 4 \): x-intercept \( (8, 0) \)
  • Vertically dilated function \( y = 3x - 24 \): x-intercept \( (8, 0) \)

Conclusion:

Both functions have the same x-intercept, which is \( (8, 0) \). The correct answer from the options provided is:

  • The original function has an x-intercept of (8,0) and the dilated function has an x-intercept of (8,0).