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# 3.3: Getting Triggy With It

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 2, Lesson 6.

## Problem 1 – A general trigonometric function

Using the Transformation Graphing app, press $Y=$ and enter the general sine function in $Y_1$,

$Y_1 = A \ ^* \ \sin(B \ ^* \ X + C) + D.$

Complete the table.

$A$ $B$ $C$ $D$ zero1 zero2 min max
1 1 0 0
4 $\frac{1}{2}$ 3 1

## Problem 2 – The effect of the coefficients A, B, C, and D

Examining A

• Set $B = 1$ and $C = D = 0$ and change the value of $A$. Try 4 different values of $A$.
$A$ $B$ $C$ $D$ zero1 zero2 min max
1 0 0
1 0 0
1 0 0
1 0 0
• How did the appearance of the graph change?
• Which graph features changed? Which did not change?
• Write equations to describe the relationship between $A$ and the features that did change.
• When $B = 1$ and $C = D = 0$, ___________________.

The value of $A$ is the amplitude. It is equal to half of the difference between its maximum and minimum values.

• Calculate the amplitude from the minimum and maximum values in the table above.
• Compare the results to the values of $A$. What do you notice?

Examining B

$A$ $B$ $C$ $D$ zero1 zero2 min max
1 0 0
1 0 0
1 0 0
1 0 0
• Try 4 different values of $B$. How did the appearance of the graph change?
• Which graph features changed? Which did not change?
• Describe the relationship between $B$ and the features that did change.

Examining C

$A$ $B$ $C$ $D$ zero1 zero2 min max
1 1 0
1 1 0
1 1 0
1 1 0
• Try 4 different values of $C$. How did the appearance of the graph change?
• Which graph features changed? Which did not change?
• What is the effect of an increasing sequence of values for $C$ on the graph?
• What is the effect of a decreasing sequence of values for $C$ on the graph?

Examining D

$A$ $B$ $C$ $D$ zero1 zero2 min max
1 1 0
1 1 0
1 1 0
1 1 0
• Try 4 different values of $D$. How did the appearance of the graph change?
• Try an increasing sequence of values for $D$ such as 0, 1, 2, 3, 4 ... What is the effect on the graph?
• Try a decreasing sequence of values for $D$ such as 0, -1, -2, -3, -4 ... What is the effect on the graph?
• Describe the effect of the value of $D$ on the graph. How does changing $D$ change the graph features?

## Problem 3 – A closer look at amplitude, period, and frequency

In $Y_1$, enter the general cosine function, $A \ ^* \ \cos(B \ ^* \ X + C) + D$.

amplitude: half of the vertical distance from minimum value to maximum value

period: horizontal distance from one peak (maximum point) to the next

frequency: number of cycles per $2\pi$ interval

• Write a formula to find the frequency $f$ given the period $p$.
• Use the formula to complete the table on the next page.
$A$ $B$ $C$ $D$ max point min point next max point amplitude period frequency
1 1 0 0 (0, 1) (3.14, -1) (6.28, 1)

$\frac{1}{2}*(1 - (-1))$

2

$6.28 - 0$

6.28

$2\pi$

1 0 0
1 0 0
1 0 0
1 0 0
1 1 0
1 1 0
1 1 0
1 1 0
1 1 0
• Based on the results in the table, determine and record each relationship:
• $A$ and the amplitude
• $B$ and the frequency
• $B$ and the period

Feb 23, 2012

Nov 04, 2014

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TI.MAT.ENG.SE.1.Trigonometry.3.3