Question

for each part, begin with the default trigonometric function f(x)=1.2 cos(1.4x)+2 sin(1.8x). (a) enable the graphs of f, f and f. you should see a similar shape in all three graphs. which one has the largest amplitude? o f o f o f which one has the smallest? o f o f o f (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 select for x > 0 and select when x < 0. what changes if you make a negative? o the graph of f appears to be reflected with respect to the x - axis. o the graph of f appears to be reflected with respect to the y - axis. o the period of f appears to change. o the graph of f appears to be translated upward. o the graph of f appears to be translated downward.

Explanation:

Step1: Find the first - derivative of \(f(x)\)

Using the chain - rule \((\cos(ax))^\prime=-a\sin(ax)\) and \((\sin(ax))^\prime = a\cos(ax)\), if \(f(x)=1.2\cos(1.4x)+2\sin(1.8x)\), then \(f^\prime(x)=1.2\times(- 1.4)\sin(1.4x)+2\times1.8\cos(1.8x)=-1.68\sin(1.4x) + 3.6\cos(1.8x)\).

Step2: Find the second - derivative of \(f(x)\)

\(f^{\prime\prime}(x)=-1.68\times1.4\cos(1.4x)-3.6\times1.8\sin(1.8x)=-2.352\cos(1.4x)-6.48\sin(1.8x)\).

Step3: Analyze the amplitudes

For \(y = A\cos(ax)+B\sin(ax)\), the amplitude \(A_{total}=\sqrt{A^{2}+B^{2}}\).
For \(f(x)=1.2\cos(1.4x)+2\sin(1.8x)\), \(A_{f}=\sqrt{1.2^{2}+2^{2}}=\sqrt{1.44 + 4}=\sqrt{5.44}\approx2.33\).
For \(f^\prime(x)=-1.68\sin(1.4x)+3.6\cos(1.8x)\), \(A_{f^\prime}=\sqrt{(-1.68)^{2}+3.6^{2}}=\sqrt{2.8224 + 12.96}=\sqrt{15.7824}\approx3.97\).
For \(f^{\prime\prime}(x)=-2.352\cos(1.4x)-6.48\sin(1.8x)\), \(A_{f^{\prime\prime}}=\sqrt{(-2.352)^{2}+(-6.48)^{2}}=\sqrt{5.531904+41.9904}=\sqrt{47.522304}\approx6.89\).
So \(f^{\prime\prime}\) has the largest amplitude and \(f\) has the smallest amplitude.

Step4: Analyze the effect of \(a\)

The function \(f(x)\) is given as a sum of cosine and sine functions. When we talk about increasing a parameter \(a\) (it's not clear which \(a\) in the original function, but assuming it affects the coefficients of the trigonometric functions), the amplitude of \(f\) is determined by the coefficients of \(\cos\) and \(\sin\) terms. Increasing the value of the coefficients of the trigonometric functions will increase the amplitude of \(f\) for both \(x>0\) and \(x < 0\).
If we make a coefficient negative, for a function \(y = A\cos(ax)+B\sin(ax)\), changing the sign of a coefficient is equivalent to a reflection about the \(x\) - axis. For example, if we change the sign of \(A\) in \(y = A\cos(ax)+B\sin(ax)\), the new function \(y=-A\cos(ax)+B\sin(ax)\) is the reflection of the original function about the \(x\) - axis.

Answer:

(a) Largest amplitude: \(f^{\prime\prime}\)
Smallest amplitude: \(f\)
(b) 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.