Question

predict #1
if we changed the 2 to a 5 in the equation, what would happen to
the graph?
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y = 2 sin 3(x - π) + 1

Explanation:

Step1: Recall the amplitude formula

The general form of a sine function is \(y = A\sin(B(x - C))+D\), where \(A\) is the amplitude.

Step2: Analyze the effect of changing \(A\)

In the original equation \(y = 2\sin(3(x-\pi)) + 1\), \(A = 2\). If we change \(2\) to \(5\), the new equation is \(y=5\sin(3(x - \pi))+1\), so \(A = 5\). The amplitude of a sine - wave \(y = A\sin(B(x - C))+D\) is given by \(|A|\). The amplitude determines the vertical stretch of the graph. A larger \(|A|\) value means a greater vertical stretch.

Answer:

The graph of the sine function would be vertically stretched. The amplitude of the sine wave, which is currently \(2\), would change to \(5\). So the distance from the mid - line (\(y = 1\)) to the maximum and minimum values of the function would increase. The mid - line \(y = 1\) (determined by \(D = 1\) in the general form \(y=A\sin(B(x - C))+D\)) remains the same, but the function would oscillate between \(y=1 + 5=6\) (maximum) and \(y=1-5=-4\) (minimum) instead of oscillating between \(y=1 + 2 = 3\) and \(y=1-2=-1\).