Learning Objectives for Section 2.3 Quadratic Functions

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Learning Objectives for Section 2.3Quadratic
Functions

• You will be able to identify and define quadratic
  functions, equations, and inequalities.
• You will be able to identify and use properties
  of quadratic functions and their graphs.
• You will be able to solve applications of
  quadratic functions.

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Quadratic Functions

• If a, b, c are real numbers with a ? 0, then the
  function
• is a ____________________________ function,
  and its graph is a _____________________________
  ___.

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Graph of a Quadratic Function

• For each quadratic function, we will identify the
  
• axis of symmetry
• vertex
• y-intercept
• x-intercept(s), if any

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Graph of a Quadratic Function

• For each quadratic function, we will also note
  the
• domain
• range

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Two Forms of the Quadratic Function

• 1) General form of a quadratic function
• 2) Vertex form of a quadratic function

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Quadratic Function in General Form

• For a quadratic function in general form

1. Axis of symmetry is
2. Vertex

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Quadratic Function in General Form

• For a quadratic function in general form

• y-intercept Set x 0 and solve for y. Or we
  can say, find f(0)(Write as an ordered pair.)
• x-intercepts Set f(x) 0 and solve for x. We
  can factor or use the Quadratic Formula to solve
  the quadratic equation.
• (Write intercepts as ordered pairs.)

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The Quadratic Formula

• To solve equations in the form of

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Vertex of a Quadratic Function

• Example
• Find axis of symmetry and vertex of
• To find the axis of symmetry
• To find the vertex

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Intercepts of a Quadratic Function

• Example
• Find the x and y intercepts of
• 1) To find the y-intercept(Write as an ordered
  pair.)

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Intercepts of a Quadratic Function (continued)

• 2) To find the x intercepts of
  (round to nearest
  tenth write as ordered pairs.)

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Graph of a Quadratic Function

• Now sketch the graph of
  

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Quadratic Function in Vertex Form
For a quadratic function in vertex form

• Vertex is (h , k)
• Axis of symmetry x h
• y-intercept Set x 0 and solve for y. Or we
  can say, find f(0)
• x-intercepts Set f(x) 0 and solve for x.

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Quadratic Function in Vertex Form
Example Find vertex and axis of symmetry
of Vertex Axis of symmetry
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Quadratic Function in Vertex Form
Example Find the intercepts of y-intercept
x-intercepts
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Quadratic Function in Vertex Form

• Now sketch the graph of
  

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Break-Even Analysis
The financial department of a company that
produces digital cameras has revenue (in millions
of dollars) and cost functions for x million
cameras as follows R(x) x(94.8 - 5x) C(x)
156 19.7x. Both have domain 1 lt x lt
15 Break-even points are the production levels at
which ________________________________________.
Use the graphing calculator to find the
break-even points to the nearest thousand
cameras.
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Graphical Solution to Break-Even Problem
1) Enter the revenue function into y1 y1 2)
Enter the cost function into y2 y2 3) In
WINDOW, change xmin1, xmax15, ymin ____, and
ymax_______. 4) Graph the two functions. 5)
Find the intersection point(s) using CALC 5
Intersection
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Solution to Break-Even Problem(continued)
Here is what it looks like if we graph the cost
and revenue functions on our calculators. You
need to find each intersection point separately.
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Solution to Break-Even Problem(continued)
Now, lets graph the PROFIT function P(x)
__________________________ Where would you
find the break-even points on the graph of the
profit function?
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Solution to Break-Even Problem(continued)
Use the graph to find the MAXIMUM PROFIT.
?? 4maximum
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Quadratic Regression
A visual inspection of the plot of a data set
might indicate that a parabola would be a better
model of the data than a straight line. In that
case, rather than using linear regression to fit
a linear model to the data, we would use
quadratic regression on a graphing calculator to
find the function of the form y ax2 bx c
that best fits the data. From the ?? CALC menu,
choose 5 QuadReg
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Example of Quadratic Regression
An automobile tire manufacturer collected the
data in the table relating tire pressure x (in
pounds per square inch) and mileage (in thousands
of miles.) x Mileage 28 45 30 52 32 55 34 51 36 47
Using quadratic regression on a graphing
calculator, find the quadratic function that best
fits the data. Round values to 6 decimal places.
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Example of Quadratic Regression(continued)
Enter the data in a graphing calculator and
obtain the lists below.
Choose quadratic regression from the statistics
menu and obtain the coefficients as shown
This means that the equation that best fits the
data is y -0.517857x2 33.292857x-
480.942857
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Example of Quadratic Regression(continued)
If appropriate, use the model to estimate the
number of miles you could get from tires inflated
at a) 35 psi and b) 50 psi.