Use a graphing calculator to graph the original function y=12x−4

Question

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 (−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 (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 (48,0)
.
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 48 comma 0 right parenthesis .

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

Answer 1

To determine the x-intercepts of the original function $y=12x-4$ and the vertically dilated function with a dilation factor $k=6$, we first find the x-intercept of the original function.

Original Function:

Set $y=0$: $$0=12x-4$$

Add 4 to both sides: $$4=12x$$

Divide by 12: $$x=\frac{4}{12}=\frac{1}{3}$$

Thus, the x-intercept of the original function is: $$(\frac{1}{3},0)$$

Vertically Dilated Function:

The dilated function can be expressed as: $$y=k\cdot (12x-4)=6(12x-4)$$ Expanding this: $$y=72x-24$$

Now, set $y=0$: $$0=72x-24$$

Add 24 to both sides: $$24=72x$$

Divide by 72: $$x=\frac{24}{72}=\frac{1}{3}$$

Thus, the x-intercept of the vertically dilated function is also: $$(\frac{1}{3},0)$$

Conclusion:

Both the original function and the vertically dilated function have the same x-intercept, which is $(\frac{1}{3},0)$.

None of the provided options correspond to this result, so it seems there may be an error in the given options.