NZ Level 8 (NZC) Level 3 (NCEA) [In development] Dilation of tangent curves
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 $a\sin x$asinx or $a\cos x$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 $\frac{2}{7}\tan x$27tanx, $\tan x$tanx, and $3\tan x$3tanx 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, $a\tan x$atanx, we can determine the value of the coefficient $a$a by comparing the values of $\tan x$tanx and $a\tan x$atanx at a particular value of $x$x. A good choice of $x$x would be $x=\frac{\pi}{4}$x=π4 since $\tan\frac{\pi}{4}=1$tanπ4=1. Then, $a\tan\frac{\pi}{4}=a$atanπ4=a.

#### Example

Determine the coefficient $a$a for the following tangent curve (the one shown in black). When $\tan x=1$tanx=1, $a\tan x=3.5$atanx=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$3.5tanx.

#### Practice questions

##### Question 1

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

##### Question 2

Consider the functions $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=3\tan x$g(x)=3tanx.

1. 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).

A

They have the same period.

B

They approach their asymptotes at the same rate.

C

They have the same asymptotes.

D

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

A

They have the same period.

B

They approach their asymptotes at the same rate.

C

They have the same asymptotes.

D

##### Question 3

Consider the right-angled triangle shown below, where the length of $OB$OB is $1$1 unit and the angle $x$x is measured in radians. 1. What is the length of the side $AB$AB? Give your answer in terms of $x$x.

2. In this image, $\triangle COD$COD is similar to $\triangle AOB$AOB, and the length of $OD$OD is $a$a units. What is the length of the side $CD$CD? Give your answer in terms of $a$a and $x$x. 3. This graph shows how the length of the side $AB$AB changes as we change the size of the angle $x$x.

If the length of $OD$OD is $2$2 units, draw the graph that shows how the length of the side $CD$CD will change as we change the size of the angle $x$x.

4. Using the applet below to change the length of $CD$CD, select the most appropriate description of the graph of $y=a\tan x$y=atanx.

Scaling the unit circle by a factor of $a$a corresponds to horizontally translating of the graph of $y=\tan x$y=tanx by $a$a units.

A

Scaling the unit circle by a factor of $a$a corresponds to vertically dilating of the graph of $y=\tan x$y=tanx by a factor of $a$a.

B

Horizontally translating the graph of $y=\tan x$y=tanx by $a$a units corresponds to a horizontal translation of the unit circle by $a$a units.

C

Vertically dilating the graph of $y=\tan x$y=tanx by a factor of $a$a corresponds to a rotation about the unit circle of $a$a radians.

D

Scaling the unit circle by a factor of $a$a corresponds to horizontally translating of the graph of $y=\tan x$y=tanx by $a$a units.

A

Scaling the unit circle by a factor of $a$a corresponds to vertically dilating of the graph of $y=\tan x$y=tanx by a factor of $a$a.

B

Horizontally translating the graph of $y=\tan x$y=tanx by $a$a units corresponds to a horizontal translation of the unit circle by $a$a units.

C

Vertically dilating the graph of $y=\tan x$y=tanx by a factor of $a$a corresponds to a rotation about the unit circle of $a$a radians.

D

### Outcomes

#### M8-2

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