About Graphs of
Functions: Examples
TABLE OF CONTENTS
Properties ............................................................................................................................ 1
Even Functions................................................................................................................ 1
Odd Functions................................................................................................................. 1
Symmetry........................................................................................................................ 1
Increasing and Decreasing Functions ............................................................................. 2
Basic Functions................................................................................................................... 2
Basic Functions............................................................................................................... 2
Polynomial Functions ......................................................................................................... 4
Factoring ......................................................................................................................... 4
Graphing ......................................................................................................................... 5
Leading Coefficients....................................................................................................... 6
Rational Functions .............................................................................................................. 7
Definition/Form .............................................................................................................. 7
Asymptotes ..................................................................................................................... 8
Defined Functions............................................................................................................... 9
Domain and Range.......................................................................................................... 9
Graphing ....................................................................................................................... 10
New Functions .................................................................................................................. 11
Shifting of Graphs......................................................................................................... 11
Combining the Shifts .................................................................................................... 12
Compression and Stretching ......................................................................................... 12
Reflection...................................................................................................................... 13
About Graphs of Functions: Examples
Properties
Even Functions
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Below are two graphs of even functions.
Odd Functions
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Below are two graphs of odd functions.
Symmetry
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Both graphs below evidence "mirror image" properties, qualifying
each as examples of symmetry.
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Increasing and Decreasing Functions
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Below are graphs of increasing and decreasing functions.
Increasing
Decreasing
Basic Functions
Basic Functions
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Below are the domain, range and a sample graph of the constant
function y = C. In this case, C = 2.
Domain: ( -∞, +∞)
Range:
{C}
2
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Below are the domain, range, and the graph of the identity function
y = x.
Domain: ( -∞, +∞)
Range: ( -∞, +∞)
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Below are the domain, range and the graph of the function y = x².
Domain: ( -∞, +∞)
Range:
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[ 0, +∞ )
Here are the domain, range and the graph of the function y = x³.
Domain: ( -∞, +∞)
Range: ( -∞, +∞)
3
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Below are the domain, range and the graph of the function y = 1/x.
Domain: ( -∞, 0 ) U ( 0, +∞)
Range: ( -∞, 0 ) U ( 0, +∞)
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Below are the domain, range and the graph of the function y = x1/3.
Domain: ( -∞, +∞ )
Range: ( -∞, +∞)
Polynomial Functions
Factoring
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20x2 + 23x - 21 = (5x - 3)(4x + 7)
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x5 + 15x4 + 90x3 +270x2 + 405x + 243 = (x + 3)5
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x3 – 4x2 – x + 4 = (x - 1)(x + 1)(x - 4)
►
x2 = (x - 0)(x - 0) = (x - 0)2
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Graphing
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Below is a degree 6 polynomial (notice the 5 bumps, as 5 + 1 = 6):
x6 + 5x5 + 6x4 - 4x3 - 8x2 = x2(x + 2)3(x - 1)
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Below is a degree 5 polynomial (notice the 4 bumps, as 4 + 1 = 5):
x5 + 5x4 + 3x3 - 9x2 = x2(x - 1)(x + 3)2
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Below is a degree 4 polynomial (notice the 3 bumps, as 3 + 1 = 4):
x4 - 5x2 + 4 = (x + 2)(x + 1)(x - 1)(x - 2)
5
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Below is a degree 3 polynomial (notice the 2 bumps, as 2 + 1 = 3):
x3 + x2 - 2x = x(x - 1)(x + 2)
Leading Coefficients
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Case 1: Consider the polynomial 3x2 + 5x - 4 (Fig 1).
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Case 2: Consider the polynomial -2x4 + 10 (Fig 2).
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Case 3: Consider the function 5x3 -7x +2 (Fig 1).
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Case 4: Consider the polynomial -2x3 + 20 (Fig 2).
Rational Functions
Definition/Form
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Below are some examples of rational functions.
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Below are some examples of rational functions in factored form.
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Below are some examples of rational functions in reduced factored
form.
Asymptotes
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The function (25x7 + 7) / (5x7 + 4x4 + 3x3) has a numerator of degree
7 and a denominator of degree 7. Since m = n, there is a horizontal
asymptote at 25/5 = 5.
8
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The function (-28x2 + 7x + 1) / (4x2 + 2x - 8) has numerator of degree
2 and a denominator of degree 2. Since m = n, there is a horizontal
asymptote at -28/4 = -7.
Note that the function crosses the asymptote on the right of the y-axis.
This occasionally occurs with horizontal asymptotes, but never with
vertical asymptotes.
Defined Functions
Domain and Range
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Consider the following function:
Domain = ( -∞, +∞ )
Range = ( -∞, 0 ) U { 1 }
9
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Consider the following function:
Domain = ( -∞, 2 )
Range = ( -∞, 0 ) U ( 0 , 2 )
Graphing
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Consider the following piecewise function:
10
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Consider the following piecewise function:
New Functions
Shifting of Graphs
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y = f(x) + c
y = f(x) = x² ( - - - line)
y = x² + 2
►
(
line)
y = f(x) + c
y = f(x) = x² ( - - - line)
y = (x - 4)²
(
line)
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Combining the Shifts
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Consider the graph below.
Compression and Stretching
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Consider the following graphs, in which c ≥ 1:
y = 2[x]
y = [2x]
12
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Consider the following graphs, in which 0 < c < 1:
y = (1/2) [x]
y = [(1/2) x]
Reflection
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Note that the two graphs below are mirror images of one another, as
the only difference in their respective formulas is that one has a
negative term.
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