Dilation of tangent curves

In an earlier chapter, the effect of multiplying a function by a constant was explained. In the case of the sine and cosine functions, we saw that the constant $a$a in $a\sin x^\circ$asinx° or $a\cos x^\circ$acosx° gave the amplitude of the function, because the maximum and minimum values of the sine and cosine functions are multiplied by $a$a.

The idea of amplitude does not apply to the tangent function because there is no maximum or minimum value. However, a similar idea of dilation in the vertical direction does apply.

In the following diagrams, the graphs of $\tan x^\circ$tanx°, $\frac{2}{7}\tan x^\circ$27​tanx° and $3\tan x^\circ$3tanx° are displayed in sequence to illustrate the effect of increasing the coefficient that multiplies the function.

The steepness of the curve near the origin increases as the coefficient increases, indicating a stretch in the vertical direction.

Given a graph that looks like a tangent function, $a\tan x^\circ$atanx°, we can determine the value of the coefficient $a$a by comparing the values of $\tan x^\circ$tanx° and $a\tan x^\circ$atanx° at a particular value of $x$x. A good choice of $x$x would be $x=45^\circ$x=45° since $\tan45^\circ=1$tan45°=1. Then, $a\tan45^\circ=a.$atan45°=a.

Example

Determine the coefficient $a$a for the following tangent curve (the one shown in black).

When $\tan x=1$tanx=1, $a\tan x=4$atanx=4. Therefore, $a=4$a=4 and the graph shown in black is the graph of the function $4\tan x^\circ$4tanx°.

Examples

QUESTION 1

QUESTION 2

QUESTION 3