Question

How do you work out where construction lines go when sketching graphs such as `f(x)=x sin x`?

I am finding it difficult to understand how to calculate construction lines for graphs such as that given above.

Here, I know the lines are at `y=abs(x)` and `y=-abs(x)` but how do I work this out for myself?

Answer 1

When you graph any function of the form `A(x)sinx` we note that as `x` varies then `sinx` oscillates between `+-1`. As a result the graph of `y=A(x)sinx` will be sinusoidal and oscillate between `+-A(x)` (the amplitude).

The same principle applies equally other periodic functions,

Thus if we want to graph `y=sinx` we first start with separate sketches of `y=sinx`
graph{sinx [-15, 15, -10, 10]}

And `y=+-x`:
graph{(y-x)(y+x)=0 [-15, 15, -10, 10]}

We can now sketch the oscillations so that they lie between the `A(x)` function:
graph{(y-x)(y+x)(y-xsinx)=0 [-15, 15, -10, 10]}

Leading to the final graph
graph{xsinx [-15, 15, -10, 10]}