A Study of Function Transformations
Name ________________________
Using the Desmos graph: https://www.desmos.com/calculator/b49yekjjy4, follow the instructions about inserting
an “a” and adding the slider. Describe how the coefficients affect the shape of the graph.
A. a  f  x 

With any function, f(x), what is the effect of multiplying the function by positive coefficient?

What is the effect of multiplying the function by a negative coefficient?
B. f  x   a

With any function, f(x), what is the effect of adding a positive constant to the function?

What is the effect of adding a negative constant (or subtracting a number) to the function?
C. f  x  a 

With any function, f(x), what is the effect of adding a positive constant to the x in the function?

What is the effect of adding a negative constant (or subtracting a number) from the x of the function?
D. f a  x 

With any function, f(x), what is the effect of multiplying the x in the function by positive coefficient
between 0 and 1?

With any function, f(x), what is the effect of multiplying the x in the function by positive coefficient
between greater than 1?

What is the effect of multiplying the x in the function by a negative coefficient?
See if these relationships hold for other functions. Click on the menu icon in the upper left corner and add a
New Blank Graph. Enter y 
x  h  k and add all sliders. Do the numbers in the square root function
effect the graph the same as in the first function? ________
h shifts the graph ____________________, k shifts the graph _______________________
Try other functions: y  a x  h  k
and
y  a x  h   k
2
How do coefficients and constants affect trigonometric graphs? You can start with a new blank graph again.
Enter the parent function sin(x) in one line and then f  x   a  sin b  x  c   d in another line. It will be
helpful to only adjust one slider at a time and always return to the parent function settings which are
A=1
B=1
C=0
D=0
f  x   1sin 1x  0  0
What does a effect in the graph? _________________________________________________________________
What does b control? __________________________________________________________________________
What does c control? ___________________________________________________________________________
What does d control? ___________________________________________________________________________
Go to wsfcs.k12.nc.us/jriggins and click on Math III on the left. Scroll down to the bottom of that page and click
on Trig data to model. Copy the data for NC Temperature and paste in a blank line in the desmos graphing
page. Using the sliders and what you have learned about the coefficients a, b, c, and d try to build a sine
function to model this data. To help us be consistent, let’s keep A and B both positive.
_______________________________________
Copy the data for ND Temperature and paste it into a blank line in the desmos screen. This time DO NOT use
the sliders. Type in what coefficients you think will model the data and adjust them as needed to fit the data.
_______________________________________
What does the difference in the amplitude (the “a” coefficient) of the two models tell you about the
temperatures in North Carolina and North Dakota?
____________________________________________________________________________________________
What does the difference in the vertical shift (the “d” constant) of the two models tell you about the
temperatures in North Carolina and North Dakota?
____________________________________________________________________________________________
The “b” coefficient controls the length of the cycle. Since the length of our cycle is 12 months we would use the
equation
2
b
 12 . Solving for b you would get b 

6
, or approximately .5
The horizontal shift of the sine function is right 4 months (the average temp in April is the average yearly temp
or the midline), Thus solving .5(4)  c  0 for c we get -2 for the c coefficient.