Question

1. which function below describes this graph?

a. $y = \sin(x - 3)$ c. $y = \sin x$
b. $y = \sin x + 3$ d. $y = 3\sin x$

1. which function below describes this graph?

a. $y = \sin x$ c. $y = \sin x + 2$
b. $y = \sin 2x$ d. $y = 2\sin x$

Explanation:

Question 29

Step1: Recall sine function properties

The standard sine function \( y = \sin x \) has an amplitude of 1, passes through the origin, and has a period of \( 2\pi \).

Step2: Analyze each option

• Option A: \( y=\sin(x - 3) \) is a horizontal shift of \( y=\sin x \) by 3 units to the right. The graph in the question passes through the origin, so this is not correct.
• Option B: \( y=\sin x+3 \) is a vertical shift of \( y = \sin x \) by 3 units up. The midline of this graph would be \( y = 3 \), but the given graph has a midline at \( y = 0 \) (passes through the origin and oscillates around the x - axis), so this is incorrect.
• Option C: \( y=\sin x \) has an amplitude of 1. Looking at the graph, the maximum value (amplitude) seems to be 3 (since the graph goes up to \( y = 3 \) and down to \( y=- 3\)), so this is incorrect.
• Option D: \( y = 3\sin x \) has an amplitude of 3 (since the amplitude of \( A\sin x \) is \( |A| \)), passes through the origin (when \( x = 0 \), \( y=0 \)), and has the same period as \( y=\sin x \). This matches the features of the given graph.

Step1: Recall sine function transformations

The general form of a sine function is \( y = A\sin(Bx)+C \), where \( A \) is the amplitude, \( B \) affects the period (\( \text{Period}=\frac{2\pi}{|B|} \)), and \( C \) is the vertical shift.

Step2: Analyze each option

• Option A: \( y=\sin x \) has a period of \( 2\pi \).
• Option B: For \( y=\sin(2x) \), the period is \( \frac{2\pi}{2}=\pi \). The graph in the question seems to have a period that is shorter than \( 2\pi \) (more oscillations in the same interval), so we consider this.
• Option C: \( y=\sin x + 2 \) is a vertical shift of \( y=\sin x \) by 2 units up. The graph in the question does not seem to be shifted vertically (it oscillates around the x - axis), so this is incorrect.
• Option D: \( y = 2\sin x \) has an amplitude of 2 and a period of \( 2\pi \), same as \( y=\sin x \) in terms of period.

Looking at the graph, it has more oscillations (a shorter period) than \( y=\sin x \). The function \( y = \sin(2x) \) has a period of \( \pi \), which means it completes a full oscillation in the interval \( [0,\pi] \) while \( y=\sin x \) completes a full oscillation in \( [0,2\pi] \). So the graph with a shorter period is likely \( y=\sin(2x) \).

Answer:

D. \( y = 3\sin x \)

Question 30