Transcript Quadratic functions
Learning Objectives for Section 2.3
Quadratic 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
ax
2
bx
c
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: 4
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:
ax
2
2) Vertex form
of a quadratic function: ) 2
k
If
a
0, then parabola opens _________________ If
a
0, then parabola opens _________________ 6
Quadratic Function in General Form
For a quadratic function in
general form
:
1. Axis of symmetry
is
x
b
2
a
ax
2
2. Vertex
:
b
2
a
,
f
b
2
a
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Quadratic Function in General Form
For a quadratic function in
general form
:
ax
2
3. y-intercept
: Set
x
= 0 and solve for
y
.
(Write as an ordered pair.) Or we can say, find
f
(0)
4. 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.) 8
The Quadratic Formula
To solve equations in the form of
ax
2
bx
0
x
b
2 4
ac
2
a
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Vertex of a Quadratic Function
Example: Find axis of symmetry and vertex of 1.
To find the
axis of symmetry
: 3
x
2 6
x
1 2.
To find the
vertex
: 10
Intercepts of a Quadratic Function
Example: Find the
x
and
y
intercepts of 1) To find the
y-intercept
:
(Write as an ordered pair.)
3
x
2 6
x
1 11
Intercepts of a Quadratic Function (continued)
2) To find the
x intercepts
3
x
2 6
x (round to nearest tenth; write as ordered pairs.)
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Graph of a Quadratic Function
3
x
6
x
1 13
Quadratic Function in Vertex Form
For a quadratic function in vertex form: ) 2
k
1. Vertex
is (h , k)
2. Axis of symmetry
:
x
=
h
3. y-intercept
: Set
x
= 0 and solve for
y
.
Or we can say, find
f
(0)
4. 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
: (
x
4) 2 5
Axis of symmetry
: 16
Quadratic Function in Vertex Form
Example: Find the intercepts of
y-intercept
: (
x
4) 2 5
x-intercepts
: 17
Quadratic Function in Vertex Form
( ) (
x
4) 2 5 18
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 - 5
x
)
C
(
x
) = 156 + 19.7
x
. Both have domain 1 <
x
< 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. 19
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 xmin=1, xmax=15, ymin= ____, and ymax=_______.
4) Graph the two functions.
5) Find the intersection point(s) using CALC 5: Intersection 20
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, let’s graph the PROFIT function P(
x
) = __________________________ Where would you find the break-even points on the graph of the profit function? 22
Solution to Break-Even Problem
(continued) Use the graph to find the MAXIMUM PROFIT.
4:maximum 23
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
calculator to find the function of the form
y
=
ax
2 on a graphing +
bx
+
c
that best fits the data. From the CALC menu, choose 5: QuadReg 24
Example of Quadratic Regression
28 30 32 34 36 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 45 52 55 51 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
= -0.517857
x
2 + 33.292857
x
- 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.
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