Transcript 1) Quadratic Formulas and Their Graphs

5.1 Modeling Data with
Quadratic Functions
1. Quadratic Functions and Their Graphs
1) Quadratic Formulas and Their
Graphs
A quadratic function is a function that produces a
parabola.
1) Quadratic Formulas and Their
Graphs
A quadratic function is a function that produces a
parabola.
1) Quadratic Formulas and Their
Graphs
A quadratic function is a function that produces a
parabola.
4
3
2
1
0
-3
-2
-1
0
-1
-2
1
2
3
1) Quadratic Formulas and Their
Graphs
The equation of a quadratic function can be written in
standard form.
f ( x)  ax  bx  c
2
Quadratic
term
Linear
term
Constant
term
1) Quadratic Formulas and Their
Graphs
Since the largest exponent of function is 2, we say that
a quadratic equation has a degree of 2.
Equations of second degree are called quadratic.
1) Quadratic Formulas and Their
Graphs
Example 1:
Determine whether each function is linear or quadratic.
Identify the quadratic term, linear term and constant term.
a) f ( x)  x( x  3)
b) f ( x)  ( x  5x)  x
2
2
1) Quadratic Formulas and Their
Graphs
Example 1:
Determine whether each function is linear or quadratic.
Identify the quadratic term, linear term and constant term.
a) f ( x)  x( x  3)
 x  3x
2
This IS a quadratic function.
QUADRATIC TERM: x2
LINEAR TERM: 3x
CONSTANT TERM: none
b) f ( x)  ( x  5x)  x
2
2
1) Quadratic Formulas and Their
Graphs
Example 1:
Determine whether each function is linear or quadratic.
Identify the quadratic term, linear term and constant term.
a) f ( x)  x( x  3)
 x  3x
2
b) f ( x )  ( x 2  5 x )  x 2
 x  5x  x
2
 5x
2
This IS a quadratic function.
This is NOT a quadratic function.
QUADRATIC TERM: x2
QUADRATIC TERM: none
LINEAR TERM: 3x
LINEAR TERM: 5x
CONSTANT TERM: none
CONSTANT TERM: none
1) Quadratic Formulas and Their
Graphs
We can graph parabolas using a table of values.
1) Quadratic Formulas and Their
Graphs
We can graph parabolas using a table of values.
Recall…graphing linear functions…
1) Quadratic Formulas and Their
Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of
values.
1) Quadratic Formulas and Their
Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of
values.
x
-2
-1
0
1
2
y
1) Quadratic Formulas and Their
Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of
values.
x
y
-2
(-2)2 = 4
-1
(-1)2 = 1
0
(0)2 = 0
1
(1)2 = 1
2
(2)2 = 4
1) Quadratic Formulas and Their
Graphs
Example 2:
Graph the parent function f(x) = x2 using a table of
values.
x
y
-2 4
-1 1
0
0
1
1
2
4
1) Quadratic Formulas and Their
Graphs
The axis of symmetry is a line
that divides the parabola in half.
1) Quadratic Formulas and Their
Graphs
The axis of symmetry is a line
that divides the parabola in half.
The vertex is a maximum or
minimum of the parabola.
1) Quadratic Formulas and Their
Graphs
The axis of symmetry here is
x=0
The vertex here is a minimum at
(0, 0)
1) Quadratic Formulas and Their
Graphs
Points on the parabola have
corresponding points that
are equidistant from the axis of
symmetry.
A
B
A’
B’
1) Quadratic Formulas and Their
Graphs
Example 3:
Identify the vertex and axis of symmetry for the parabola. Identify
points corresponding to P and Q.
4
33
P
22
11
0
-2
-2
-1
-1
0
-1
-1
-2
-2
11
22Q
3
3
44
1) Quadratic Formulas and Their
Graphs
Example 3:
Identify the vertex and axis of symmetry for each parabola.
Identify points corresponding to P and Q.
4
P
33
Vertex: (1, -1)
P’
P
22
Axis of symmetry: x = 1
11
P’ (3, 3)
0
-2
-2
-1
-1
Q’ 0
-1
-1
-2
-2
11
22Q
3
3
44
Q’ (0, 0)
Fitting a Quadratic Function to 3
Points
Find a quadratic function that includes the values (2,3),
(3,13) and (4,29).
Fitting a Quadratic Function to 3
Points
Find a quadratic function whit a graph that includes
(1,0), (2,-3), (3,-10)