Question

what does changing the value of a do to our graph? what key features are impacted by this change? what happens when a is negative?

Explanation:

When considering the general form of a sine or cosine function \( y = A\sin(Bx - C)+D \) or \( y=A\cos(Bx - C)+D \), the value of \( A \) is called the amplitude.

• **Effect of changing \( A \) (when \( A

eq0 \))**:

• The amplitude of the sine or cosine function is \( |A| \). So, changing the value of \( A \) changes the amplitude of the graph. The amplitude determines the maximum distance of the graph from the mid - line \( y = D \). For example, if we have \( y=\sin(x) \) (\( A = 1 \)), the graph oscillates between \( y=- 1\) and \( y = 1\). If we have \( y = 2\sin(x) \) (\( A=2 \)), the graph oscillates between \( y=-2\) and \( y = 2\).
• The key feature impacted is the amplitude. The period (\( T=\frac{2\pi}{|B|} \)), phase shift (\( \frac{C}{B} \)) and vertical shift (\( D \)) are not affected by the value of \( A \) (assuming \( B\), \( C\), and \( D\) remain constant).
• Effect when \( A\) is negative:
• A negative \( A\) reflects the graph of the sine or cosine function over the \( x\) - axis. For example, if \( y=\sin(x) \) has a graph that goes up from the origin for \( x\in(0,\frac{\pi}{2})\), then \( y =-\sin(x) \) has a graph that goes down from the origin for \( x\in(0,\frac{\pi}{2})\). The amplitude \( |A| \) still determines the maximum distance from the mid - line. So, if \( y=-3\cos(x)\), the amplitude is \( |- 3|=3 \) and the graph of \( y = \cos(x) \) is reflected over the \( x\) - axis.

Answer:

Changing \( A \) changes the amplitude of the sine or cosine graph. The key feature impacted is the amplitude. When \( A\) is negative, the graph of the sine or cosine function is reflected over the \( x\) - axis while still having an amplitude of \( |A| \).