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

2. 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

3. Introduction to Writing Trigonometric Equations

4. 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!

5. Writing Equations When Amplitude is Modified

6. 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:

7. 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 :

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

9. Writing Equations When a Vertical Translation is Applied

10. 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

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

12. Writing Equations When Period is Modified

13. 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:

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

15. Writing Equations When a Phase Shift is Applied

16. 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

17. 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:

18. Writing Equations When Phase Shift is Modified (Example)
Ex 4: Write an equation to match the graph – assume the period is 2π:

19. Writing Equations in General

20. 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

21. 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!

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

23. 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)