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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

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**Introduction to Writing Trigonometric Equations**

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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**

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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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**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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**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**

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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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**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**

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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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**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**

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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

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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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**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**

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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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**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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4.5 – Graphs of Sine and Cosine A function is periodic if f(x + np) = f(x) for every x in the domain of f, every integer n, and some positive number p.

4.5 – Graphs of Sine and Cosine A function is periodic if f(x + np) = f(x) for every x in the domain of f, every integer n, and some positive number p.

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