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Finding an Equation from Its Graph Trigonometry MATH 103 S. Rook

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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 2

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Introduction to Writing Trigonometric Equations

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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

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Writing Equations When Amplitude is Modified

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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: 6

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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 : 7

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Writing Equations When Amplitude is Modified (Example) Ex 1: Find an equation to match the graph: a) b) 8

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Writing Equations When a Vertical Translation is Applied

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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 10

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Writing Equations When a Vertical Translation is Applied (Example) Ex 2: Write an equation to match the graph: 11

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Writing Equations When Period is Modified

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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: 13

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Writing Equations When Period is Modified (Example) Ex 3: Write an equation to match the graph: 14

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Writing Equations When a Phase Shift is Applied

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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 max 2 Decreases to middle 3 Decreases to min 4 Rises to middle 5 Rises to max 16

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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: 17

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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: 18

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Writing Equations in General

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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 20

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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

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Writing Equations in General Ex 5: Write an equation to match the graph: a) b) 22

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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) 23

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