Trigonometric Graphs

New Zealand

Level 7 - NCEA Level 2

Lesson

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 $a\sin x^\circ$`a``s``i``n``x`° or $a\cos x^\circ$`a``c``o``s``x`° 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 $\tan x^\circ$`t``a``n``x`°, $\frac{2}{7}\tan x^\circ$27`t``a``n``x`° and $3\tan x^\circ$3`t``a``n``x`° 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, $a\tan x^\circ$`a``t``a``n``x`°, we can determine the value of the coefficient $a$`a` by comparing the values of $\tan x^\circ$`t``a``n``x`° and $a\tan x^\circ$`a``t``a``n``x`° at a particular value of $x$`x`. A good choice of $x$`x` would be $x=45^\circ$`x`=45° since $\tan45^\circ=1$`t``a``n`45°=1. Then, $a\tan45^\circ=a.$`a``t``a``n`45°=`a`.

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

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

Choose the description that best matches the graph of $y=5\tan x$`y`=5`t``a``n``x`.

The graph of $y=\tan x$

`y`=`t``a``n``x`has been compressed vertically.AThe graph of $y=\tan x$

`y`=`t``a``n``x`has been stretched vertically.BThe graph of $y=\tan x$

`y`=`t``a``n``x`has been stretched vertically and reflected across the $x$`x`-axis.CThe graph of $y=\tan x$

`y`=`t``a``n``x`has been compressed vertically and reflected across the $x$`x`-axis.D

Each of the graphs shown below are of equations in the form $y=a\tan x$`y`=`a``t``a``n``x`. For which graph is $-1$−1$<$<$a$`a`$<$<$0$0$?$?

- Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Consider the graph of $y=a\tan x$`y`=`a``t``a``n``x` shown below.

Loading Graph...

What vertical dilation factor would we need to apply to the graph of $y=\tan x$

`y`=`t``a``n``x`to obtain this graph?