1. Digital Lesson
Graphs of Functions

2. Steps to Graph a Function
The graph of a function y = f (x) is a set of ordered pairs (x, f (x)), for values of x in the domain of f.
To graph a function:
1. Make a table of values.
2. Plot the points.
3. Connect them with a curve.
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Steps to Graph a Function

3. Example: Graph a Function
Example: Graph the function f (x) = x2 – 2x – 2. x y x
f (x) = x2 – 2x – 2 y
(x, y) -2
f (-2) = (-2)2 – 2(-2) – 2 = 6 6
(-2, 6) -1
f (-1) = (-1)2 – 2(-1) – 2 = 1 1
(-1, 1)
f (0) = (0)2 – 2(-0) – 2 = -2
(0, -2)
f (1) = (1)2 – 2(1) – 2 = -3 -3
(1, -3) 2
f (2) = (2)2 – 2(2) – 2 = -2
(2, -2) 3
f (3) = (3)2 – 2(3) – 2 = 1
(3, 1) 4
f (4) = (4)2 – 2(4) – 2 = 6
(4, 6)
(4, 6)
(-2, 6)
(0, -2)
(-1, 1)
(3, 1)
(1, -3)
(2, -2)
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Example: Graph a Function

4. The domain is the real numbers.
The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. x y 4 -4
Example: Find the domain and range of f (x) = x2 – 2x – 2 by investigating its graph.
Range
The domain is the real numbers.
The range is { y: y ≥ -3} or [-3, + ].
Domain = all real numbers
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Domain & Range

5. Example: Domain & Range
Example: Find the domain and range of the function f (x) = from its graph.
The graph is the upper branch of a parabola with vertex at (-3, 0). x y
Range
(-3, 0)
Domain
The domain is [-3,+∞] or {x: x ≥ -3}.
The range is [0,+∞] or {y: y ≥ 0}.
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Example: Domain & Range

6. Example: The following equations define relations: y 2 = x y = x 2
A relation is a correspondence that associates values of x with values of y.
The graph of a relation is the set of ordered pairs (x, y) for which the relation holds.
Example: The following equations define relations:
y 2 = x
y = x 2
x 2 + y 2 = 4 x y
(4, 2)
(4, -2) x y x y
(0, 2)
(0, -2)
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Relation

7. Vertical Line Test
A relation is a function if no vertical line intersects its graph in more than one point.
Of the relations y 2 = x, y = x 2, and x 2 + y 2 = 1 only y = x2 is a function.
Consider the graphs.
y 2 = x x y
y = x 2 x y
x 2 + y 2 = 1 x y
2 points of intersection
1 point of intersection
2 points of intersection
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Vertical Line Test

8. Example: Vertical Line Test
Vertical Line Test: Apply the vertical line test to determine which of the relations are functions. y
x = | y – 2| y
x = 2y – 1 x x
The graph does not pass the vertical line test. It is not a function.
The graph passes the vertical line test. It is a function.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Vertical Line Test