Question

(b) if you increase the value of a, what is the visible effect on the graph of f for x > 0? for x < 0? when the value of a increases, the amplitude of f tends to increase for x > 0 and increase when x < 0. what changes if you make a negative? the graph of f appears to be reflected with respect to the x - axis. the graph of f appears to be reflected with respect to the y - axis. the period of f appears to change. the graph of f appears to be translated upward. the graph of f appears to be translated downward. does varying the value of a have a similar effect on the graphs of f and f? yes no (c) if you vary the value of b, does the amplitude of the graph of f change? yes no does the period? yes no what about in the graph of f? the graph of f takes on similar changes. the changes to the graph of f are drastically different than the changes to f. the graph of f shows no changes when b is varied.

Explanation:

1. When \(a\) is made negative in a function \(y = a\cdot f(x)\), the sign of the \(y -\)values changes. This results in a reflection of the graph of \(y = f(x)\) across the \(x -\)axis.
2. For the derivatives \(f'\) and \(f''\), varying \(a\) in \(y = a\cdot f(x)\) just multiplies \(f(x)\) by a constant. The derivative of \(y=af(x)\) is \(y'=a\cdot f'(x)\) and the second - derivative is \(y'' = a\cdot f''(x)\). So varying \(a\) has a similar effect (scaling) on \(f'\) and \(f''\) as on \(f\).
3. The amplitude of a function of the form \(y = a\sin(bx)+c\) (or similar trig - functions) is determined by \(|a|\), not \(b\). So varying \(b\) does not change the amplitude. The period of a function \(y=\sin(bx)\) is \(T=\frac{2\pi}{|b|}\), so varying \(b\) changes the period. When considering the derivative of a trig - function like \(y = \sin(bx)\) (where \(y'=b\cos(bx)\)), changing \(b\) also affects the derivative's period and other characteristics in a way that is related to the changes in the original function.

Answer:

1. The graph of \(f\) appears to be reflected with respect to the \(x\) - axis.
2. Yes
3. No
4. Yes
5. The graph of \(f'\) takes on similar changes.