finding an equation from its graph trigonometry math 103 s. rook
TRANSCRIPT
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
• 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
• 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
• 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
• 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
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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
• 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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