practice testMultiple Choice

Practice Test

Question 4 • Math 3

Which of the following is the graph of y is equal to sine of open paren 4 times open paren x minus pi close paren close paren ?

Answer

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis. The vertical axis is labeled from negative 2 to 2 in increments of 1. The horizontal axis has tick marks labeled pi and 2 pi to the right of the origin, and negative pi and negative 2 pi to the left of the origin. A wave curve with a peak at 1 and a trough at negative 1 passes through 0 comma 0. To the right of the origin, the curve has a peak between 0 comma 0 and pi, a trough between pi and 2 pi, and another peak after 2 pi before ending with a downward-pointing arrow. To the left of the origin, the curve has a trough between 0 comma 0 and negative pi, a peak between negative pi and negative 2 pi, and another trough after negative 2 pi before ending with an upward-pointing arrow.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis. The vertical axis is labeled from negative 2 to 2 in increments of 1. The horizontal axis has two tick marks to the right labeled pi and 2 pi, and one unlabeled tick mark to the left. A wave curve with a peak at 1 and a trough at negative 1 forms a W-shaped curve between each consecutive tick mark. Each W-shaped segment has two troughs, one peak in the middle, and its outer two peaks aligned directly above the tick marks. The curve crosses the vertical axis at 0 comma 1 and ends with an upward-pointing arrow on both the far right and far left.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis. The vertical axis is labeled from negative 2 to 2 in increments of 1. The horizontal axis has two tick marks to the right labeled pi and 2 pi, and one unlabeled tick mark to the left. A wave curve with a peak at 1 and a trough at negative 1 passes through 0 comma 0, completing 2 full cycles between each tick mark on the horizontal axis. The curve ends with an upward-pointing arrow just past 2 pi on the right, and a downward-pointing arrow near the tick mark on the left.

Image with description A coordinate plane with a horizontal x-axis and a vertical y-axis. The vertical axis is labeled from negative 2 to 2 in increments of 1. The horizontal axis has tick marks labeled pi and 2 pi to the right of the origin, and negative pi and negative 2 pi to the left of the origin. A wave curve with a peak at 1 and a trough at negative 1 passes through the origin 0 comma 0. The curve has one peak and one trough between each consecutive tick mark on the horizontal axis. The curve ends with an upward-pointing arrow past 2 pi to the right of the origin, and a downward-pointing arrow past negative 2 pi to the left of the origin.

Explanation

Using the identity sin(θ − 2π) = sin θ, we get sin(4(x − π)) = sin(4x − 4π) = sin(4x). Its amplitude is 1, and its period is 2π/4 = π/2, so it completes two full cycles in each interval of length π. It also passes through (0, 0), matching choice C, the graph with two cycles between consecutive π-spaced ticks.

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