To find the x-intercepts of the functions \( y = 12x - 4 \) and its vertically dilated version, we first need to determine the x-intercept of the original function.
Finding the x-intercept of the original function:
Set \( y = 0 \):
\[ 0 = 12x - 4 \]
Add 4 to both sides:
\[ 4 = 12x \]
Now, divide both sides by 12:
\[ x = \frac{4}{12} = \frac{1}{3} \]
Thus, the x-intercept of the original function is \( \left( \frac{1}{3}, 0 \right) \).
Finding the vertically dilated function:
The vertically dilated function with a factor \( k = 6 \) is given by multiplying the original function by \( 6 \):
\[ y = 6(12x - 4) = 72x - 24 \]
Finding the x-intercept of the dilated function:
Set \( y = 0 \):
\[ 0 = 72x - 24 \]
Add 24 to both sides:
\[ 24 = 72x \]
Now, divide both sides by 72:
\[ x = \frac{24}{72} = \frac{1}{3} \]
Conclusion on the x-intercepts:
The x-intercepts are:
- For the original function: \( \left( \frac{1}{3}, 0 \right) \)
- For the dilated function: \( \left( \frac{1}{3}, 0 \right) \)
Response:
The correct option is:
The original function has an x-intercept of \( ( \frac{1}{3}, 0 ) \), and the dilated function has an x-intercept of \( ( \frac{1}{3}, 0 ) \).