Question

the functions y = sin(x) and y = sin(ax) + b, for constants a and b, are graphs in the standard (x,y) coordinate plane below. one of the following statements about a and b is true. which statement is it? choose the option that best answers the question. 0 < a < 1 and b < 0 a > 1 and b < 0 0 < a < 1 and b > 0 a > 1 and b > 0

Explanation:

Step1: Analyze the period of the function \(y = \sin(ax)+b\)

The period of the function \(y=\sin(x)\) is \(2\pi\). For the function \(y = \sin(ax)\), the period is \(T=\frac{2\pi}{|a|}\). From the graph, the period of \(y = \sin(ax)+b\) is less than \(2\pi\) (since the original \(y = \sin(x)\) has a period of \(2\pi\) and the transformed sine - like function oscillates faster). If \(T=\frac{2\pi}{|a|}<2\pi\), then \(|a|> 1\). Since \(a\) is a non - zero real number (as it is a coefficient in the sine function transformation), and assuming \(a>0\) (because the basic shape of the sine function is preserved in terms of increasing and decreasing intervals direction), we have \(a > 1\).

Step2: Analyze the vertical shift \(b\)

The mid - line of \(y=\sin(x)\) is \(y = 0\). The mid - line of \(y=\sin(ax)+b\) is above \(y = 0\). For a sine function \(y = A\sin(ax)+b\), the mid - line is \(y = b\). Since the mid - line of \(y=\sin(ax)+b\) is above \(y = 0\), we have \(b>0\).

Answer:

\(a > 1\) and \(b>0\)