4.3-4 Quadratic Functions and Models
I. Definition
A quadratic function is a function of the form
(standard form) where a, b, and c are real numbers and a  0.
The domain of a quadratic function consists of all real numbers
unless it is in an application problem that dictates otherwise.
II. Graphing a Quadratic Function Using Transformations
A quadratic function in transformation form
is the parabola y  ax 2 shifted horizontally h units and vertically k units.
As a result, the vertex is (h, k).
The graph opens up if a > 0 and down if a < 0.
The axis of symmetry is the vertical line x = h.
However, most quadratics are not given in this form.
Getting them into this form requires completing the square, which is tedious.
EX. Graph
a. Does the parabola open up or down?
b. Find the vertex.
c. Axis of symmetry
d. Use symmetry and plot more points as needed to draw the graph.
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III. Identify the Vertex and Axis of Symmetry of a Quadratic Function
where a, b, c, are real numbers and a  0
Let
Shape of graph: Parabola
Domain: all real numbers
If a › 0 parabola opens up
If a ‹ 0 parabola opens down
The absolute value of a tells you about any vertical stretch or compression.
Vertex: (
(
))
maximum or minimum point on the parabola
Axis of symmetry: vertical line passing through vertex:
b
2a
x-intercepts: Use the discriminant to determine the # of x-intercepts
no x-intercepts
one x-intercept (touches the x-axis at the vertex)
has two distinct x-intercepts: cross x-axis twice
y – intercept
EX.
2
e. Does the parabola open up or down?
f. Find the vertex.
g. Axis of symmetry
h. x-intercept(s)
i. y-intercept
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EX.
a. Does the parabola open up or down?
b. Find the vertex.
c. Axis of symmetry
d. x-intercept(s)
e. y-intercept
EX.
a. Does the parabola open up or down?
b. Find the vertex.
c. Axis of symmetry
d. x-intercept(s)
e. y-intercept
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EX.
a. Does the parabola open up or down?
b. Find the vertex.
c. Axis of symmetry
d. x-intercept(s)
e. y-intercept
IV. Finding the Maximum or minimum value of a quadratic function
Demand Equation The price p in dollars and the quantity x sold of a certain product
obey the demand equation


Revenue = # items produced ∙ price of each item
Express the revenue R as a function of x.
Find the number of items to sell to maximize the revenue.
Find the maximum revenue.
Find the price to charge for maximum revenue.
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Ex: Suppose that the manufacturer of a washing machine has found that when the unit
price is p dollars, the revenue R is
a. What unit price should be established for the washing machine to maximize revenue?
b. What is the maximum revenue?
EX: Enclosing a Rectangular Field along a River: 3000 feet of fencing are available to
enclose a rectangular field. No fencing will be used on the side along the river.
a.
Express the area A of the rectangle as a function of the width w of the rectangle.
b. For what dimensions will the area be the largest?
c.
What is the maximum area?
V. Fitting a Quadratic Function to Data
Quadratic data will have scatter plots that suggest a parabolic model:
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Fitting a Quadratic Function to Data
The method is the same as for linear regression, except you choose quadratic
regression:
Press STAT, then under the CALC menu, choose 5:QuadReg , ENTER.
To graph the curve:
Press Y= VARS choose 5:Statistics; then under the EQ menu, choose 1:RegEQ.
Ex: Automobile Speed for Optimal Gas Mileage
An engineer collects the following data for a Toyota Camry. Speed s is in miles per hour,
and gas mileage is in miles per gallon.
Avg. Speed
30
35
40
40
45
50
55
60
65
65
70
Gas Mileage
18
20
23
25
25
28
30
29
26
25
25
a. Create a scatter plot. Does the data appear to be linear or quadratic in nature?
b. Find the equation of the best fitting model. Round to 3 decimal places.
c. Use the model to estimate the speed that will yield the best gas mileage, rounded to
the nearest .1 mi/hour.
d. According to the model, what is the best gas mileage for this car, rounded to the
nearest .1 mi/gal?
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