Chapter 2
Functions and
Graphs
Section 3
Quadratic Functions
Learning Objectives for Section 2.3
Quadratic Functions
 The student will be able to identify and define quadratic
functions, equations, and inequalities.
 The student will be able to identify and use properties of
quadratic functions and their graphs.
 The student will be able to solve applications of quadratic
functions.
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Quadratic Functions
If a, b, c are real numbers with a not equal to zero, then
the function
f ( x)  ax  bx  c
2
is a quadratic function and its graph is a parabola.
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Vertex Form of the
Quadratic Function
It is convenient to convert the general form of a
quadratic equation
f ( x)  ax  bx  c
2
to what is known as the vertex form:
f ( x)  a( x  h)  k
2
This is done by completing the square which will be
reviewed in a few slides.
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Generalization
For







f ( x)  a( x  h)2  k
Vertex is at (h , k)
If a > 0, the graph opens upward.
If a < 0, the graph opens downward.
Axis of symmetry: x = h
k is the minimum if a > 0, otherwise its the maximum
Domain: −∞, ∞
Range:
• If a < 0  −∞, 𝑘
• If a > 0  𝑘, ∞
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Generalization
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General Form to Vertex Form
Completing the Square
The example below illustrates the procedure:
Consider 𝑓 𝑥 = 3𝑥 2 − 6𝑥 − 1
Complete the square to find the vertex.
f (x) = (3x2 – 6x) –1
Group first two terms
f (x) = 3(x2 – 2x) –1
Factor out coef. of x2
f (x) = 3(x2 – 2x +1) –1 – 3 Complete the square
inside the parentheses
Since you’re really adding 3, you have to subtract 3
f (x) = 3(x – 1)2 – 4
Vertex (1, -4); opens upwards
Axis of sym: x = 1; Minimum = -4
D: (-,  ); R: [-4, )
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Example
 Rewrite the function in vertex form:
𝑓 𝑥 = −𝑥 2 + 8𝑥 − 9
𝑓 𝑥 = −𝑥 2 + 8𝑥 − 9
𝑓 𝑥 = − 𝑥 2 − 8𝑥 − 9
𝑓 𝑥 = − 𝑥 2 − 8𝑥 + 16 − 9 + 16
𝑓 𝑥 =− 𝑥−4
2
+7
Vertex (4, 7); opens downwards
Axis of sym: x = 4; Maximum = 7
D: (-,  ); R: (-, 7]
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Intercepts
 Y-intercept
• Plug in x = 0
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Intercepts
 Find the y intercept of:
f ( x)  3x  6 x  1
2
f (0)  3(0)  6(0)  1
2
y − intercept is: −1
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Intercepts
 X-intercepts
• It might have 0, 1, or 2 x-intercepts
• They can be determined by:
o Factoring (if possible)
o Completing the square
o Quadratic Formula
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Intercepts
 Find the x intercepts of 𝑓 𝑥 = 𝑥 2 + 5𝑥 − 14
0= 𝑥+7 𝑥−2
𝑥+7=0
𝑥 = −7
𝑥−2=0
𝑥=2
The x − intercepts are: −7 and 2.
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Intercepts
 Find the x intercepts of f ( x)  3x
 Using the quadratic formula:
2
 6x 1
−6 ± 62 − 4 −3 −1
𝑥=
2 −3
−6 ± 24
=
−6
−6 ± 2 6
=
−6
Exact
−3 ± 6
=
−3
Barnett/Ziegler/Byleen Business Calculus 12e
Approx.
≈ 1.82, 0.18
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Finding x-intercepts
Using a Graphing Calculator
• Graph: y = –x2 + 5x + 3
• Select CALC (2nd Trace)
• Select 2: zero
• Left bound? Use arrows to position cursor
to the left of intercept, then hit ENTER.
• Right bound? Use arrows to position cursor
to the right of intercept, then hit ENTER.
• Guess? Hit ENTER.
• Zero  -0.5414
• Repeat to find other zero.
• Zero  5.5414
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Max and Min Values
 A parabola that opens upwards has a minimum value.
 A parabola that opens downwards has a maximum value.
 In either case, the max/min value is the y-coordinate of the
vertex.
 Finding the vertex from the equation:
• 𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 +𝑘  Vertex (h, k)
• 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐  Vertex
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−𝑏
2𝑎 , 𝑓
−𝑏
2𝑎
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Example
 Find the maximum or minimum of each function:
𝑓 𝑥 =− 𝑥+3
2
+7
𝑉𝑒𝑟𝑡𝑒𝑥 −3, 7
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑜𝑓 7.
𝑓 𝑥 = 3𝑥 2 − 12𝑥 + 14
𝑏
−12
−
=−
=2
2𝑎
6
𝑓 2 =2
𝑉𝑒𝑟𝑡𝑒𝑥 2, 2
𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑜𝑓 2
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Finding Max/Min Using Graphing
Calculator
• Graph: y = –x2 + 5x + 3
• Select CALC (2nd Trace)
• Select 4: maximum
• Left bound?
• Right bound?
• Guess?
• Maximum 9.25
• Vertex: (2.5, 9.25)
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Quadratic Inequalities
 Two Methods for Solving Quadratic Inequalities
• Algebraic
o Do not need to review this procedure yet.
• Graphing Calculator
o This is the procedure we will use for now.
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Quadratic Inequalities
1. Graph the function.
2. Determine its x-intercepts using the Calc Zero function.
3. If inequality is:
1. f(x) > 0 then state the intervals for which the graph is
above the x-axis.
2. f(x) < 0 then state the intervals for which the graph is
below the x-axis.
3. f(x)  0 then state the intervals for which the graph is
on or above the x-axis.
4. f(x)  0 then state the intervals for which the graph is
on or below the x-axis.
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Solving Quadratic Inequalities
Solve the quadratic inequality –x2 + 5x + 3 > 0 .
x-intercepts are: -0.5414 and 5.5414
The graph is on or above the
x-axis over the interval:
[– 0.5414, 5.5414 ]
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Solving Quadratic Inequalities
Solve the quadratic inequality –x2 + 5x + 3 < 0 .
x-intercepts are: -0.5414 and 5.5414
The graph is below the x-axis over
the interval:
−∞, – 0.5414)  (5.5414 , ∞)
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Applications
 There are many applications involving quadratic functions.
 Let’s look at an example…
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Break-Even Analysis
The financial department of a company that produces digital
cameras has the revenue and cost functions for x million
cameras as follows:
R(x) = x(94.8 – 5x)
C(x) = 156 + 19.7x. Both have domain 1 < x < 15
Break-even points are the production levels at which
R(x) = C(x). Find the break-even points (using your
graphing calculator) to the nearest million cameras.
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Solution to Break-Even Problem
(continued)
If we graph the cost and revenue functions on a
graphing utility, we obtain the following graphs,
showing the two intersection points:
x = 2.490 or 12.530
The company breaks even when they sell
approximately 2 million cameras and 13 million
cameras.
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Quadratic Regression
Evinrude Outboard Motor
Speed
(mph)
Miles per
Gallon
10.3
4.1
18.3
5.6
24.6
6.6
29.1
6.4
33.0
6.1
36.0
5.4
38.9
4.9
•
•
•
•
•
•
•
Insert speed in L1 and mpg in L2.
Turn Plot 1 on
ZoomStat
STAT
CALC
5:QuadReg
Regression equation is:
• 𝑦 = −.00968𝑥 2 + 0.5069𝑥 − 0.1794
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