Trigonometric Graphs
Name _______________
Date ________________
In order to make graphing sine curves more efficient, it’s helpful to learn how the sine
curve changes when the constant values a, b, c, and d change in the function:
y = d + asin(bx - c)
We will take a look at how each constant changes the sine curve on an individual basis.
USING YOUR GRAPHING CALCULATOR, ENTER Y = SIN(X) INTO Y1 AND LEAVE IT
THERE FOR THE DURATION OF THIS ACTIVITY. THIS IS CONSIDERED “THE PARENT
FUNCTION”.
I.
Amplitude
The a value in this function y = sin(x) is 1.
Enter the following functions into your graphing calculator: y = 2sin(x), y = 4sin(x),
and y = 0.5sin(x) separately and compare with the parent function.
a. Does the amplitude of the original sine curve change?
b. How does the a value change the original sine curve?
The AMPLITUDE of y = asin(x) represents half the distance
between the maximum and minimum values of the function. It
is represented by a .
In other words, when the amplitude or a is 1 (like in the original), the maximum value
is 1 and the minimum value is -1. But for our graph of y = 2sin(x), the maximum value
will now be 2 and the minimum value will now be -2. Half the distance between these
two values being 2.
Trigonometric Graphs
II.
Name _______________
Date ________________
Period
You should still have y = sin(x) in Y1. The a value is 1 and the b value, which
we will investigate now, is also 1.
Enter the function y = sin(2x) into your graphing calculator. How does it compare to
the parent function?
The PERIOD of a trigonometric function is the length of the
piece of the function that is repeated throughout the domain.
Let b be a real positive number. The period of y = asin(bx) is
given by
2
b

For what values of b will the sine curve appear to be stretched out?
For what values of b will the sine curve appear to be compressed?
III.
Shifts
A. Enter y = sin(x) into Y=.
Now enter y = 1+ sin(x) and graph both functions.
a. How did the graph change from the parent function?
b. Replace y = 1 + sin(x) with y = -1 + sin(x). How did the graph change
from the parent function?
Trigonometric Graphs
Name _______________
Date ________________
c. How does the constant value d change the sine curve from the parent
function?
B. Leave y = sin(x) in Y1. Enter the function y = sin(x + 1) into your calculator
and graph.
a. How did the new graph change from the parent function?
b. Replace y = sin(x + 1) with y = sin(x – 1). How is this new function
different than the parent function?
c. How does the constant value c change the sine curve from the parent
function?
IV.
Practice
Describe how each function below will change from the parent function y = sin(x).
1. y = sin (x – 2)
2. y = sin(x + 3)
3. y = 2 + sin(x – 4)
4. y = 2sin(x)
5. y = sin(4x)
6. y = 1 + sin(2x)
7. y = 3sin(x – 4)
8. y = -3 + sin(2x – 1)
9. y = -1 +2sin(4x – 2)