Quadratic Functions
Copyright © Cengage Learning. All rights reserved.
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4.1
Quadratic Functions and Parabolas
Copyright © Cengage Learning. All rights reserved.
Objectives
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Recognize a quadratic from its graph and
equation.
Identify when a quadratic graph is increasing or
decreasing.
Identify the vertex of a parabola and explain its
meaning.
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Introduction to Quadratics and
Identifying the Vertex
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Introduction to Quadratics and Identifying the Vertex
Let us consider the functions
T(m) = –1(m – 8)2 + 72 and B(t) = 28.7t2 – 286.2t + 6899.3
These functions are not linear, since they have a squared
term and cannot be written in the general form of a line,
Ax + By = C.
These functions are instead called quadratic functions.
Quadratic functions are most commonly represented by
either the standard form or the vertex form.
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Introduction to Quadratics and Identifying the Vertex
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Introduction to Quadratics and Identifying the Vertex
Parabolas can be described as a “U shape” or an
“upside-down U shape.” A parabola will change from
increasing to decreasing or from decreasing to increasing.
The point where the parabola turns around and goes in the
opposite direction is called the vertex.
The vertex represents the point where the lowest or
minimum value occurs on the graph if the graph opens
upward and represents the highest or maximum value on
the graph if the graph opens downward.
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Introduction to Quadratics and Identifying the Vertex
Examples of each of these types of graphs are shown
below. Data that show this pattern can be modeled by
using a quadratic function.
Parabola opens upward.
Vertex: (4, 1)
Lowest point
Minimum point
Decreasing when x < 4
Increasing when x > 4
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Introduction to Quadratics and Identifying the Vertex
Parabola opens downward.
Vertex: (5, 4)
Highest point
Maximum point
Increasing when x < 5
Decreasing when x > 5
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Introduction to Quadratics and Identifying the Vertex
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Recognizing Graphs of Quadratic
Functions and Identifying the Vertex
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Example 1 – Reading a quadratic graph
Use the graph of f(x) to estimate the following.
a. For what x-values is this curve increasing? Decreasing?
Write your answer as inequalities.
b. Vertex
c. x-intercept(s)
d. y-intercept
e. f(5) = ?
f. What x-value(s)
will make f(x) = –2?
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Example 1 – Solution
a. Reading the graph from left to right, we see that the
curve is increasing for x < 2 and decreasing for x > 2.
b. This curve changes from increasing to decreasing when
x = 2, so the vertex is (2, 3).
c. The curve crosses the x-axis at x = 0 and x = 4, so
(0, 0) and (4, 0) are the x-intercept(s).
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Example 1 – Solution
cont’d
d. The curve crosses the y-axis at y = 0, so the y-intercept
is (0, 0).
e. When x = 5, the curve has an output of about y = –3.5,
so f(5) = –3.5.
f. The output of the function is y = –2 when the input is
about x = –0.5 and x = –4.5.
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