Describe how the graphs of y + 3 = 2sin (3x + 90°) compares to the graph of y = sin (x).
Is my answer correct?
y = 2 sin [3(x + 30)] - 3
amplitude = 2 (compared to 1 for y = sin x)
period = 360 / 3 = 120 (compared to 360 for y = sin x)
phase shift = -30 (compared to 0 for y = sin x)
center of oscillation = -3 (compared to 0 for y = sin x)
the given graph is stretched vertically by 2
shrunk horizontally by a factor of 3 (3 cycles where sin x has 1)
shifted 30 degrees to the left
shifted down 3
perfect.
To compare the graphs of y + 3 = 2sin (3x + 90°) and y = sin (x), let's break it down step by step.
Step 1: Starting with the standard graph of y = sin (x), which represents a basic sine wave that oscillates between -1 and 1.
Step 2: In y + 3 = 2sin (3x + 90°), we have a few modifications to the equation. Firstly, there is an added constant of 3 on the y-axis, shifting the entire graph vertically upward by 3 units.
Step 3: Next, the coefficient of the sine function is 2, leading to a change in amplitude. The amplitude of y + 3 = 2sin (3x + 90°) is twice that of y = sin (x), so the peaks and troughs will be magnified.
Step 4: Finally, inside the sine function, we have 3x + 90°. This indicates a horizontal compression by a factor of 1/3 and a phase shift of -90 degrees to the left.
Combining all the modifications, the graph of y + 3 = 2sin (3x + 90°) will be similar to the graph of y = sin (x), except it will be shifted vertically upward by 3 units, have peaks and troughs magnified by a factor of 2, compressed horizontally by a factor of 1/3, and shifted 90 degrees to the left.
To describe how the graphs of y + 3 = 2sin (3x + 90°) and y = sin (x) compare, we need to understand the effects of the given transformations on the original graph of y = sin (x). Let's break it down step by step.
1. Amplitude:
In the equation y + 3 = 2sin (3x + 90°), the 2 in front of the sin function represents the amplitude. The original amplitude of y = sin (x) is 1. When multiplied by 2, it increases to 2. This means that the peaks and troughs of the graph will be stretched vertically by a factor of 2.
2. Horizontal Stretch:
The coefficient of 3 inside the sin function, in the form 2sin (3x + 90°), represents the horizontal stretch factor. To find the period of the function, we divide 2π (the period of sin (x)) by the number inside the sin function, giving us a period of 2π/3. This means that the graph will complete one full cycle in a shorter distance, resulting in a tighter curve.
3. Phase Shift:
The angle of 90° added inside the sin function, as seen in 2sin (3x + 90°), represents a phase shift. This means that the graph will shift 90° to the left. In other words, the entire graph will be shifted to the right by one-quarter of its period.
4. Vertical Shift:
The "+ 3" in the equation y + 3 = 2sin (3x + 90°) represents a vertical shift. Since we add 3 to the entire sin function, the graph will shift upward by 3 units.
Combining all of these transformations, the graph of y + 3 = 2sin (3x + 90°) will have a larger amplitude, a shorter horizontal period, a phase shift to the right by one-quarter of its period, and a vertical shift upward by 3 units compared to the graph of y = sin (x).