Trigonometric Graphs

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 $a\sin x$`a``s``i``n``x` or $a\cos x$`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 $\frac{2}{7}\tan x$27`t``a``n``x`, $\tan x$`t``a``n``x`, and $3\tan x$3`t``a``n``x` are displayed on the same set of axes 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, $a\tan x$`a``t``a``n``x`, we can determine the value of the coefficient $a$`a` by comparing the values of $\tan x$`t``a``n``x` and $a\tan x$`a``t``a``n``x` at a particular value of $x$`x`. A good choice of $x$`x` would be $x=\frac{\pi}{4}$`x`=π4 since $\tan\frac{\pi}{4}=1$`t``a``n`π4=1. Then, $a\tan\frac{\pi}{4}=a$`a``t``a``n`π4=`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, $a\tan x=3.5$`a``t``a``n``x`=3.5. Therefore, $a=3.5$`a`=3.5 and the graph shown in black is the graph of the function $3.5\tan x$3.5`t``a``n``x`.

The graph of $y=\tan x$`y`=`t``a``n``x` is shown below. On the same set of axes, draw the graph of $y=5\tan x$`y`=5`t``a``n``x`.

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Consider the functions $f\left(x\right)=\tan x$`f`(`x`)=`t``a``n``x` and $g\left(x\right)=3\tan x$`g`(`x`)=3`t``a``n``x`.

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What do the graphs of these functions have in common? Select all options that apply.

They both pass through the point $\left(\frac{\pi}{4},1\right)$(π4,1).

AThey have the same period.

BThey approach their asymptotes at the same rate.

CThey have the same asymptotes.

DThey both pass through the point $\left(\frac{\pi}{4},1\right)$(π4,1).

AThey have the same period.

BThey approach their asymptotes at the same rate.

CThey have the same asymptotes.

D

Consider the right-angled triangle shown below, where the length of $OB$`O``B` is $1$1 unit and the angle $x$`x` is measured in radians.

What is the length of the side $AB$

`A``B`? Give your answer in terms of $x$`x`.In this image, $\triangle COD$△

`C``O``D`is similar to $\triangle AOB$△`A``O``B`, and the length of $OD$`O``D`is $a$`a`units. What is the length of the side $CD$`C``D`? Give your answer in terms of $a$`a`and $x$`x`.This graph shows how the length of the side $AB$

`A``B`changes as we change the size of the angle $x$`x`.If the length of $OD$

`O``D`is $2$2 units, draw the graph that shows how the length of the side $CD$`C``D`will change as we change the size of the angle $x$`x`.Loading Graph...Using the applet below to change the length of $CD$

`C``D`, select the most appropriate description of the graph of $y=a\tan x$`y`=`a``t``a``n``x`.Scaling the unit circle by a factor of $a$

`a`corresponds to horizontally translating of the graph of $y=\tan x$`y`=`t``a``n``x`by $a$`a`units.AScaling the unit circle by a factor of $a$

`a`corresponds to vertically dilating of the graph of $y=\tan x$`y`=`t``a``n``x`by a factor of $a$`a`.BHorizontally translating the graph of $y=\tan x$

`y`=`t``a``n``x`by $a$`a`units corresponds to a horizontal translation of the unit circle by $a$`a`units.CVertically dilating the graph of $y=\tan x$

`y`=`t``a``n``x`by a factor of $a$`a`corresponds to a rotation about the unit circle of $a$`a`radians.DScaling the unit circle by a factor of $a$

`a`corresponds to horizontally translating of the graph of $y=\tan x$`y`=`t``a``n``x`by $a$`a`units.AScaling the unit circle by a factor of $a$

`a`corresponds to vertically dilating of the graph of $y=\tan x$`y`=`t``a``n``x`by a factor of $a$`a`.BHorizontally translating the graph of $y=\tan x$

`y`=`t``a``n``x`by $a$`a`units corresponds to a horizontal translation of the unit circle by $a$`a`units.CVertically dilating the graph of $y=\tan x$

`y`=`t``a``n``x`by a factor of $a$`a`corresponds to a rotation about the unit circle of $a$`a`radians.D

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions