Question
A graph labeled f left-parenthesis x right-parenthesis equals 6 pi sine left-parenthesis 4 x right-parenthesis is shown from negative pi to pi.The graph completes one full cycle over the interval 0 to Start Fraction pi over 2 End Fraction. It increases from 0 to 6 pi, and then decreases passing the x-axis at x equals Start Fraction pi over 4 End Fraction, down to negative 6 pi then increases back up to the x-axis at Start Fraction pi over 2 End Fraction. This pattern continues to the left and to the right.
Question
Use the graph to answer the question.
What set describes the zeroes of the function f left parenthesis x right parenthesis equals 6 pi sine left parenthesis 4 x right parenthesis shown in the graph?
(1 point)
Responses
left brace x vertical line x equals k pi right brace for every integer k
Image with alt text: left brace x vertical line x equals k pi right brace for every integer k
left brace x vertical line x equals 4 k pi right brace for every integer k
Image with alt text: left brace x vertical line x equals 4 k pi right brace for every integer k
left brace x vertical line x equals Start Fraction k over 4 End Fraction pi right brace for every integer k
Image with alt text: left brace x vertical line x equals Start Fraction k over 4 End Fraction pi right brace for every integer k
left brace x vertical line x equals 6 k right brace for every integer k
Image with alt text: left brace x vertical line x equals 6 k right brace for every integer k
Question
Use the graph to answer the question.
What set describes the zeroes of the function f left parenthesis x right parenthesis equals 6 pi sine left parenthesis 4 x right parenthesis shown in the graph?
(1 point)
Responses
left brace x vertical line x equals k pi right brace for every integer k
Image with alt text: left brace x vertical line x equals k pi right brace for every integer k
left brace x vertical line x equals 4 k pi right brace for every integer k
Image with alt text: left brace x vertical line x equals 4 k pi right brace for every integer k
left brace x vertical line x equals Start Fraction k over 4 End Fraction pi right brace for every integer k
Image with alt text: left brace x vertical line x equals Start Fraction k over 4 End Fraction pi right brace for every integer k
left brace x vertical line x equals 6 k right brace for every integer k
Image with alt text: left brace x vertical line x equals 6 k right brace for every integer k
Answers
GPT-4o mini
To find the zeros of the function \( f(x) = 6\pi \sin(4x) \), we need to solve \( 6\pi \sin(4x) = 0 \).
This equation will be zero whenever \( \sin(4x) = 0 \). The sine function is zero at integer multiples of \( \pi \):
\[
4x = k\pi
\]
for every integer \( k \).
Solving for \( x \), we get:
\[
x = \frac{k\pi}{4}
\]
Thus, the set that describes the zeros of the function \( f(x) = 6 \pi \sin(4x) \) is:
\[
\{ x \mid x = \frac{k}{4}\pi \text{ for every integer } k \}
\]
Therefore, the correct response is:
\[
\text{left brace } x \text{ vertical line } x = \frac{k}{4} \pi \text{ right brace \ for every integer } k
\]
This equation will be zero whenever \( \sin(4x) = 0 \). The sine function is zero at integer multiples of \( \pi \):
\[
4x = k\pi
\]
for every integer \( k \).
Solving for \( x \), we get:
\[
x = \frac{k\pi}{4}
\]
Thus, the set that describes the zeros of the function \( f(x) = 6 \pi \sin(4x) \) is:
\[
\{ x \mid x = \frac{k}{4}\pi \text{ for every integer } k \}
\]
Therefore, the correct response is:
\[
\text{left brace } x \text{ vertical line } x = \frac{k}{4} \pi \text{ right brace \ for every integer } k
\]