Warm Up

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• Warm Up
• 1. Evaluate x2 5x for x 4 and x 3.

36 6
2. Generate ordered pairs for the function y
x2 2 with the given domain.
D 2, 1, 0, 1, 2
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9-1
Identifying Quadratic Functions
Holt Algebra 1
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The function y x2 is shown in the graph. Notice
that the graph is not linear. A quadratic
function is any function that can be written in
the standard form y ax2 bx c, where a, b,
and c are real numbers and a ? 0. The function
y x2 can be written as y 1x2 0x 0, where
a 1, b 0, and c 0.
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Notice that the quadratic function y x2 has
constant second differences.
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Example 1A Identifying Quadratic Functions
Tell whether the function is quadratic. Explain.
Since you are given a table of ordered pairs with
a constant change in x-values, see if the second
differences are constant.






x
y
2
9
7 1 1 7
1 1 1 1
6 0 6
1
2
0
1
Find the first differences, then find the second
differences.
1
0
2
7
The function is not quadratic. The second
differences are not constant.
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Example 1B Identifying Quadratic Functions
Tell whether the function is quadratic. Explain.
Since you are given an equation, use y ax2 bx
c.
y 7x 3
This is not a quadratic function because the
value of a is 0.
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Example 1C Identifying Quadratic Functions
Tell whether the function is quadratic. Explain.
y 10x2 9
Try to write the function in the form y ax2
bx c by solving for y. Add 10x2 to both sides.
This is a quadratic function because it can be
written in the form y ax2 bx c where a
10, b 0, and c 9.
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The graph of a quadratic function is a curve
called a parabola.
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Example 2B Graphing Quadratic Functions by Using
a Table of Values
Use a table of values to graph the quadratic
function.
y 4x2






Make a table of values. Choose values of x
and use them to find values of y.
Graph the points. Then connect the points with a
smooth curve.
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Check It Out! Example 2a
Use a table of values to graph each quadratic
function.
y x2 2
Make a table of values. Choose values of x
and use them to find values of y.






Graph the points. Then connect the points with a
smooth curve.
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When a quadratic function is written in the form
y ax2 bx c, the value of a determines the
direction a parabola opens.

• A parabola opens upward when a gt 0.

• A parabola opens downward when a lt 0.

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Example 3A Identifying the Direction of a
Parabola
Tell whether the graph of the quadratic function
opens upward or downward. Explain.
Identify the value of a.
Since a gt 0, the parabola opens upward.
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Example 3B Identifying the Direction of a
Parabola
Tell whether the graph of the quadratic function
opens upward or downward. Explain.
y 5x 3x2
Write the function in the form y ax2 bx c.
y 3x2 5x
a 3
Identify the value of a.
Since a lt 0, the parabola opens downward.
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The highest or lowest point on a parabola is the
vertex. If a parabola opens upward, the
vertex is the lowest point. If a parabola opens
downward, the vertex is the highest point.
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Identify the vertex of each parabola. Then give
the minimum or maximum value of the function.
The vertex is (3, 2), and the minimum is 2.
The vertex is (2, 5), and the maximum is 5.
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Check It Out! Example 4
Identify the vertex of each parabola. Then give
the minimum or maximum value of the function.
The vertex is (3, 1), and the minimum is 1.
The vertex is (2, 5) and the maximum is 5.
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For the graph of y x2 4x 5, the range
begins at the minimum value of the function,
where y 1. All the y-values of the function are
greater than or equal to 1. So the range is y ? 1.
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Example 5 Finding Domain and Range
Find the domain and range.
Step 1 The graph opens downward, so identify the
maximum.
The vertex is (5, 3), so the maximum is 3.
Step 2 Find the domain and range.
D all real numbers R y 3
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3.
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Lesson Quiz Part I
1. Is y x 1 quadratic? Explain. 2.
Graph y 1.5x2.
No there is no x2-term, so a 0.
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Lesson Quiz Part II
Use the graph for Problems 3-5. 3. Identify the
vertex. 4. Does the function have a minimum
or maximum? What is it? 5. Find the domain and
range.
(5, 4)
max 4
D all real numbers R y 4