Finding an Equation from Its Graph

TrigonometryMATH 103

S. Rook

Overview

• Section 4.5 in the textbook:– Introduction to Writing Trigonometric Equations– Writing equations when amplitude is modified– Writing equations when a vertical translation is

applied– Writing equations when period is modified– Writing equations when a phase shift is applied– Writing equations in general

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Introduction to Writing Trigonometric Equations

Introduction to Writing Trigonometric Equations

• We will only be concerned about finding equations of sine and cosine graphs

• We start with the basic graphs of y = sin x or y = cos x and then “build them up” to y = k + A sin(Bx + C) or y = k + A cos(Bx + C) – i.e. We reference each change on the given graph

to either y = sin x or y = cos x

• Finding equations from graphs can be difficult so you MUST PRACTICE!

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Writing Equations When Amplitude is Modified

Writing Equations When Amplitude is Modified

• If the minimum value m and maximum value M of the graph are values OTHER THAN -1 and 1 respectively:– The amplitude

has possibly been modified

– Calculate the value of A:

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mMA 2

1

Writing Equations When Amplitude is Modified (Continued)– If the shape of the graph appears to be flipped

“upside down” when compared to y = sin x or y = cos x:• The graph has been

reflected over the x-axis• Calculate the value

of A :

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aa

mMA 2

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Writing Equations When Amplitude is Modified (Example)

Ex 1: Find an equation to match the graph:

a) b)

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Writing Equations When a Vertical Translation is Applied

Writing Equations When a Vertical Translation is Applied

• If the minimum value m DOES NOT match the opposite of the maximum value M:– A vertical translation has been applied– Find the amplitude: – Calculate k = M – |A|

• |A| represents where the graph would normally be

• If M > |A|:– The graph was shifted up and k is positive

• If M < |A|– The graph was shifted down and k is negative

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mMA 2

1

Writing Equations When a Vertical Translation is Applied (Example)

Ex 2: Write an equation to match the graph:

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Writing Equations When Period is Modified

Writing Equations When Period is Modified

• If the graph DOES NOT have a period of 2π:– The period has been modified– Find the period• How long it takes

for the graph to complete 1 cycle

– Recall the formula for period:

– With a little algebra:

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BP

2

PB

2

Writing Equations When Period is Modified (Example)

Ex 3: Write an equation to match the graph:

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Writing Equations When a Phase Shift is Applied

Structure of the Sine and Cosine Graphs

• The sine graph has the following structure:1 Starts at middle 2 Rises to max 3 Decreases to middle 4 Decreases to min 5 Rises to middle

• The cosine graph has the following structure:1 Starts at max2 Decreases to middle 3 Decreases to min4 Rises to middle 5 Rises to max

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Writing Equations When a Phase Shift is Applied

• If the graph DOES NOT have one of these structures starting at x = 0:– A phase shift has been

applied– Find the value where

a sine or cosine period begins • Remember the

structure of each– Recall the formula to calculate phase shift: – With a little algebra:

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B

Cp

BpC

Writing Equations When Phase Shift is Modified (Example)

Ex 4: Write an equation to match the graph – assume the period is 2π:

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Writing Equations in General

Writing Equations in General

• To write an equation for a graph in general:– Take ONE step at a time– Decide whether the graph more closely resembles y

= sin x or y = cos x– Calculate:

• The value of A by utilizing the amplitude– If the graph is reflected over the x-axis, A will be negative

• The vertical translation k• The value of B by utilizing the period• The value of C by utilizing the phase shift

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Writing Equations in General (Continued)

– Write the equation of the graph as either y = k + A sin(Bx + C) or y = k + A cos(Bx + C)

• Often, there is more than one correct equation– Usually, one equation is more easier to find than

the others

• You can always check your answer by using a graphing calculator!

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Writing Equations in General

Ex 5: Write an equation to match the graph:

a) b)

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Summary

• After studying these slides, you should be able to:– Find the equation in the form of y = k + A sin(Bx + C) or y = k + A cos(Bx + C) by examining a graph

• Additional Practice– See the list of suggested problems for 4.5

• Next lesson– Inverse Trigonometric Functions (Section 4.7)

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