What's a fast way to sketch $y= \sin (5x) + \frac{3}{10}x \sin (5x)$ ?
I had this question as part of a recent test paper and I had to sketch it for the range $0 \leq x \leq \pi$ . It was for only 2 marks so there must be a faster way to sketch it than my method of serial substitution of x values.
Start by factoring the function:
$$y=\left(1+\frac 3{10}x\right)\sin(5x)$$
Putting this form together with the limits, you have a sine wave whose amplitude is linearly increasing from $1$ to $1+{3\pi\over 10}\approx 2$ and whose period is $5$ times shorter than $\sin x$. The limits are $0$ to $\pi$ and so there would be $\dfrac 52$ cycles of $2\pi$ portions of sine wave in your graph, or three waves above $0$ and two waves below, with the amplitude linearly increasing in both directions. The start and end points would both be on $y=0$, and there would be four other zeroes in between that are equally spaced across $[0,\pi]$ at $\frac \pi 5, \frac {2\pi}5,\frac {3\pi}5,\frac {4\pi}5.$