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Finding an Equation from Its
Graph
Trigonometry
MATH 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!
4
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:
1
A M m
2
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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 :
1
A  M m
2
aa
7
Writing Equations When Amplitude
is Modified (Example)
Ex 1: Find an equation to match the graph:
a)
b)
8
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: A  1 M  m
2
– 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
10
Writing Equations When a Vertical
Translation is Applied (Example)
Ex 2: Write an equation to match the graph:
11
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: P  2
B
– With a little
algebra: B  2
P
13
Writing Equations When Period is
Modified (Example)
Ex 3: Write an equation to match the graph:
14
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 max
2 Decreases to middle
3 Decreases to min
4 Rises to middle
5 Rises to max
16
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: C   Bp
p
C
B
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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!
21
Writing Equations in General
Ex 5: Write an equation to match the graph:
a)
b)
22
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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