Stretching the Tangent

Inside the bracket, b squeezes the tangent like the sine: for b positive the period of $y = \tan\left(b\cdot x\right)$ is π/b. $y = \tan\left(2x\right)$ repeats every π/2, with asymptotes at ±π/4.

In front, a stretches it upwards. The tangent has no top, so it has no amplitude; instead watch the quarter point, a quarter period right of the centre. On $y = \tan\left(x\right)$ that is (π/4, 1); on $y = 2\tan\left(x\right)$ it is (π/4, 2). A negative a turns every branch into a falling one.

On the board the tangent has two dots: the centre and the quarter point. Their gap is a quarter period across and a up.