Download Notes on Linear Functions

Notes on Transformations of Linear Functions
Name_________________________
Standards Addressed:
MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic
techniques.
a. Represent functions using function notation.
n
b. Graph the basic functions f(x) = x , where n = 1 to 3, f(x) = x, f(x) = |x|, and f(x) = x1.
c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and
y-axes.
--------------------------------------------------------------------------------------------------------------------A linear function is a function whose graph is a line.
An equation for a linear function can have x in it, but x or a quantity involving x WILL NOT...
1.
be raised to an exponent.
2.
appear in the denominator of a fraction.
3.
appear under a radical sign or a square root sign.
4.
appear inside of absolute value bars.
Instead, x may have a coefficient. Coefficients are like the “bodyguards” of variables. “If
you’re gettin’ to x, you have to go through me first!” For example, in the expression 3x, the
number ‘3’ is the coefficient of x.
In addition, numbers may be added to or subtracted from the x quantity. For example,
f ( x)  x  4 is a linear function because ‘4’ is simply subtracted from x.
Other examples of linear functions include: f ( x)  3 x 
2
1
, g ( x)  x  4 , h( x)  5 x
3
2
--------------------------------------------------------------------------------------------------------------------Are the following functions linear functions?
1.
f ( x)  10 x
2.
g ( x)  x 2  1
3.
h( x )  x 
10
7
4.
j ( x) 
1
x2
5.
m( x)  5x
6.
n( x ) 
3
x  110
4
---------------------------------------------------------------------------------------------------------------------
In the study of functions, there are a handful of parent functions. These functions are called
“parents” because many other functions come from or descend from them.
The linear parent function is f ( x)  x .
Every other linear function is a transformation of the linear parent function. As one might
suspect, the graph of f ( x)  x is a line. It is graphed below:
6
4
2
-5
5
-2
-4
-6
--------------------------------------------------------------------------------------------------------------------On the coordinate planes below, the linear parent function has already been graphed. Graph the
other function with the given equation, and then explain how the parent function's graph is
transformed to get the other function's graph.
1.
f ( x)  x
2.
g ( x)  x  4
f ( x)  x
3.
h( x )  x  2
f ( x)  x
j ( x)  3 x
6
6
6
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
f ( x)  x
4.
5.
1
k ( x)  x
4
f ( x)  x
6.
m( x )   x
f ( x)  x
n( x)  5 x
6
6
6
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
--------------------------------------------------------------------------------------------------------------------Here is a summary of all of the transformations to the parent linear function that occur by
changing the equation:
f ( x)  ax  b
If b is a positive number, then the graph shifts ________________________________________.
If b is a negative number, then the graph shifts________________________________________.
If a is a negative number, then the graph ____________________________________________.
If a  1 , then the graph is ______________________________________.
If a  1 , then the graph is ______________________________________.
--------------------------------------------------------------------------------------------------------------------Homework with Transformations of Linear Functions Name_________________________
For these examples, describe the transformation in relation to the linear parent function by
observing the equation for the function.
E1.
a ( x)  4 x
E2.
1
b( x )  x  5
3
E3.
c( x) 
5
x 9
3
--------------------------------------------------------------------------------------------------------------------Is the function a linear function?
2.
g ( x)  x 2  1
3.
h( x )   x
x3
2
5.
k ( x)  x
6.
l ( x)  4 x  10
2
x  0.3
5
8.
n( x)  5x  3
9.
p( x)  ( x  8)3
1.
f ( x)  2 x
4.
j ( x) 
7.
m( x ) 
For Questions 10-21, describe the transformation in relation to the linear parent function by
either observing the equation for the function or the graph of the function.
3
x
2
10.
r ( x) 
13.
1
u ( x)   x
2
16.
11.
s ( x)  9 x  2
12.
t ( x)   x  11
14.
v( x)  8 x  1
15.
w( x)  
17.
10
x  7.7
3
18.
6
6
6
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
19.
20.
21.
6
6
6
4
4
4
2
2
2
-5
5
-5
5
-5
5
-2
-2
-2
-4
-4
-4
-6
-6
-6
Answers:
E1. reflection; vertical stretch E2. vertical shrink; vertical shift up 5 E3. vertical stretch; vertical shift down 9
1. yes 2. no 3. yes 4. yes 5. yes 6. no 7. yes 8. no 9. no 10. vertical stretch 11. vertical stretch; vertical shift down 2
12. reflection; vertical shift up 11 13. reflection; vertical shrink 14. reflection; vertical stretch; vertical shift down 1
15. reflection; vertical stretch; vertical shift up 7.7 16. vertical shift down 4 17. vertical stretch 18. vertical shrink; vertical shift up 4
19. reflection 20. reflection; vertical stretch; vertical shift down 3 21. reflection; vertical shift up 3