Digital Lesson
Graphs of Functions
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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Example: Graph the function f (x) = x2 – 2x – 2.
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)
0
f (0) = (0)2 – 2(-0) – 2 = -2
-2
(0, -2)
1
f (1) = (1)2 – 2(1) – 2 = -3
-3
(1, -3)
2
f (2) = (2)2 – 2(2) – 2 = -2
-2
(2, -2)
3
f (3) = (3)2 – 2(3) – 2 = 1
1
(3, 1)
4
f (4) = (4)2 – 2(4) – 2 = 6
6
(4, 6)
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y
(-2, 6)
(-1, 1)
(0, -2)
(4, 6)
(3, 1)
x
(2, -2)
(1, -3)
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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.
y
Example: Find the domain and
Range
range of f (x) = x2 – 2x – 2 by
investigating its graph.
The domain is the real numbers.
The range is { y: y ≥ -3}
or [-3, + ∞].
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4
x
-4
Domain = all real numbers
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Example: Find the domain and range of the
function f (x) = x  3 from its graph.
The graph is the upper branch of a parabola with
vertex at (-3, 0).
y
Range (-3, 0)
x
Domain
The domain is [-3,+∞] or {x: x ≥ -3}.
The range is [0,+∞] or {y: y ≥ 0}.
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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 = x2
x2 + y2 = 4
y2 = x
y
y
y
(4, 2)
(0, 2)
x
(4, -2)
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x
x
(0, -2)
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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
y
y
y =x
y = x2
2
x
2 points
of intersection
x2 + y2 = 1
x
x
1 point
of intersection
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2 points
of intersection
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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
The graph does not pass
the vertical line test.
It is not a function.
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x
The graph passes the
vertical line test.
It is a function.
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The zeros of a function y = f (x) are the values of x
for which y = 0.
Example: The zeros of f (x) = 4 + 3x – x2 can be found algebraically
by setting f (x) = 0 and solving for x.
y y = 4 + 3x – x2
0 = 4 + 3x – x2 Set f (x) = 0.
(4 – x)(1 + x)
Factor.
x = -1, 4
Solve for x.
2
(4, 0)
(-1,
0)
2
The zeros of f (x) = 4 + 3x – x
x
2
are x = -1 and x = 4.
The zeros of f (x) = 4 + 3x – x2 can also be found geometrically by
observing where the graph intersects the x-axis.
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A function is:
● increasing on an interval if, for every x1 and x2 in the
interval, if x1 < x2, then f (x1) < f (x2)
● decreasing on an interval if, for every x1 and x2 in the
interval, if x1 < x2, then f (x1) > f (x2)
● constant on an interval if, for every x1
and x2 in the interval, f (x1) = f (x2).
The graph of y = f (x):
(-3, 6)
y
● Increases on [-∞, -3]
x
● Decreases on [-3, 3]
● Increases on [3, +∞].
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(3, -4)
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A function is even if f (-x) = f (x) for every x in the
domain of f.
A function is odd if f (-x) = -f (x) for every x in the
domain of f.
A function f (x) = x2 is even function,
since f (-x) = (-x)2 = x2 = f (x).
A function f (x) = x3 is an odd function,
since f (-x) = (-x)3 = -x3 = -f (x).
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A function is even if and only if its graph has line
symmetry in the y-axis.
The graphs shown have line symmetry in the y-axis.
y
y
x
f (x) = x2
y
x
f (x) = |x|
x
f (x) = cos(x)
These functions are even.
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A function is odd if and only if its graph has
point symmetry through the origin.
The graphs shown have point symmetry through
the origin.
y
y
x
f (x) = x3
y
x
f (x) = sin(x)
x
f (x) =
3
x
These functions are odd.
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