Lesson 4 Off on a Tangent Develop Understanding

Ready

The equation of a parent function is given. Write a new equation with the given transformations. Then sketch the new function on the same graph as the parent function. (If the function has asymptotes, sketch them in.)

1.

$y = x^{2}$

vertical shift: up $8$
horizontal shift: left $3$
vertical stretch or shrink: $\frac{1}{4}$

Equation:

Graph:

2.

$y = \frac{1}{x}$

vertical shift: up $4$
horizontal shift: right $3$
vertical stretch or shrink: $- 1$

a.

Equation:

b.

3.

$y = \sqrt{x}$

vertical shift: none
horizontal shift: left $5$
vertical stretch or shrink: $3$

a.

Equation:

b.

4.

$y = \sin x$

vertical shift: $1$
horizontal shift: left $\frac{π}{2}$
vertical stretch or shrink (amplitude): $3$

a.

Equation:

b.

Set

5.

Triangle $A B C$ is a right triangle. $A B = 1$ .

Use the information in the figure to label the length of the sides and measure of the angles.

6.

Triangle $R S T$ is an equilateral triangle. $R S = 1$ and $\overset{―}{S A}$ is an altitude.

Use the information in the figure to label the length of the sides, the length of $\overset{―}{R A}$ , and the exact length of $\overset{―}{S A}$ .

Label the measure of angles $R S A$ and $S R A$ .

7.

Use what you know about the unit circle and the information from the figures in problems 5 and 6 to fill in the table.

function	$θ = \frac{π}{6}$	$θ = \frac{π}{4}$	$θ = \frac{π}{3}$	$θ = \frac{π}{2}$	$θ = π$	$θ = \frac{3 π}{2}$	$θ = 2 π$
$\sin θ$
$\cos θ$
$\tan θ$

8.

Label all of the points and angles of rotation in the unit circle.

9.

Fill in the chart for $f (θ) = \tan θ$ .

$θ = \frac{π}{6}$	$θ = - \frac{π}{6}$	$θ = \frac{π}{3}$	$θ = - \frac{π}{3}$	$θ = \frac{π}{4}$	$θ = - \frac{π}{4}$

10.

Explain how the answers in problem 9 support the statement that $f (θ) = \tan θ$ is classified as an odd function.

Go

Answer the questions. Be sure you can justify your thinking.

11.

Given triangle $A B C$ with angle $C$ being a right angle, what is $m ∠ A + m ∠ B$ ?

12.

Identify the quadrants in which $\sin θ$ is positive.

13.

Identify the quadrants in which $\cos θ$ is negative.

14.

Identify the quadrants in which $\tan θ$ is positive.

15.

Explain why it is impossible for $\cos θ > 1$ .

16.

Name the angles of rotation (in radians) for when $\sin θ = \cos θ$ .

17.

For which trigonometric function does a positive rotation and a negative rotation always give the same value?

18.

Explain why in the unit circle, $\tan θ = \frac{y}{x}$ .

19.

Explain why $\sin θ = \cos (90 ° - θ)$ .