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 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 (0,−4)
, and the dilated function has an x-intercept of (0,−24)
.
The original function has an x -intercept of left parenthesis 0 comma negative 4 right parenthesis , and the dilated function has an x -intercept of left parenthesis 0 comma negative 24 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 (−8,0)
, and the dilated function has an x-intercept of (−8,0)
.

Answer 1

To find the x-intercepts of the original function \( y = 12x - 4 \) and the vertically dilated function, we first need to identify the dilated function. A vertical dilation by a factor of \( k = 6 \) modifies the original function as follows:

\[ y = k(12x - 4) \] \[ y = 6(12x - 4) = 72x - 24 \]

Next, we find the x-intercepts of both functions by setting \( y = 0 \) and solving for \( x \).

1. For the original function \( y = 12x - 4 \): \[ 0 = 12x - 4 \] \[ 12x = 4 \] \[ x = \frac{4}{12} = \frac{1}{3} \] The x-intercept is \( \left(\frac{1}{3}, 0\right) \).

2. For the vertically dilated function \( y = 72x - 24 \): \[ 0 = 72x - 24 \] \[ 72x = 24 \] \[ x = \frac{24}{72} = \frac{1}{3} \] The x-intercept is also \( \left(\frac{1}{3}, 0\right) \).

Based on the calculations:

• The original function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).
• The dilated function has an x-intercept of \( \left(\frac{1}{3}, 0\right) \).

However, none of the provided responses match this result. If I made an error in the step to determine the dilated function or x-intercepts, please advise, and I'll be happy to re-evaluate.

Answer 2

thats wrong

Answer 3

Let's properly analyze the functions step by step to find the x-intercepts.

Original Function

The original function given is: \[ y = 12x - 4 \]

To find the x-intercept, we set \( y = 0 \): \[ 0 = 12x - 4 \] \[ 12x = 4 \] \[ x = \frac{4}{12} = \frac{1}{3} \]

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

Vertically Dilated Function

The vertically dilated function with a factor of \( k = 6 \) can be written as: \[ y = 6(12x - 4) = 72x - 24 \]

Again, we find the x-intercept by setting \( y = 0 \): \[ 0 = 72x - 24 \] \[ 72x = 24 \] \[ x = \frac{24}{72} = \frac{1}{3} \]

Thus, the x-intercept of the dilated function is also: \[ \left(\frac{1}{3}, 0\right) \]

Summary of X-Intercepts

Both functions:

• Original function \( y = 12x - 4 \): x-intercept \( \left(\frac{1}{3}, 0\right) \)
• Vertically dilated function \( y = 72x - 24 \): x-intercept \( \left(\frac{1}{3}, 0\right) \)

Conclusion

Therefore, the x-intercepts of both functions are the same: \( \left(\frac{1}{3}, 0\right) \).

Since the original selected responses do not contain my answer, it seems there may be some misunderstanding about the question. If so, please clarify, and I’ll assist you further!