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2.1
Quadratic Functions and Models
Homework for section 2.1
p132
7-12 (all), 21-33 (eoo), 3545, 49-55, 65, 71, 73, 76, 81, 84
The last three word problems are
important; I will check to make sure you did
them - penalties will be assessed as needed.
We are going to be looking at graphs of polynomial
functions.
Let n be a nonnegative integer and let
a n , a n- 1, a n- 2 ,......a 2 , a 1, a 0 a ll be real numbers.
f ( x )  a n x n  a n 1x n 1  ...  a 2 x 2  a 1x 1  a 0
Called a polynomial function with degree n.
If n=0, then you have a constant function.
If n=1, then you have a linear function.
y = 3
y = 4x+3
If n=2, then you have a quadratic function.
Quadratic functions can look like any of these:
f (x)  x
2
g( x )  3  x  5
 3x  5
x2
h( x )  5 
3
j( x )   x  5
 x

2
i( x )  8 x 2  17
3

f ( x )  ax 2  bx  c
I t's graph is a parabola.
4
f ( x )  ax 2  bx  c
,
a  0
5
Y
4
3
2
f (x)  x
1
2
X
-5 -4 -3 -2 -1 0
-1
1 2
3 4 5
-2
-3
-4
-5
f ( x )  ax 2  bx  c
,
a  0
5
Y
4
3
2
f ( x )  x 2
1
-5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
X
1 2
3 4 5
10
Quadratic equations
have an:
axis of symmetry
Y
9
8
X = 6
0
-2
7
6
5
4
3
2
1
X
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9 10
10
Quadratic equations
have:
vertices, or in the
singular: a vertex
Y
9
8
7
6
5
4
3
2
1
X
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
Vertex is a minimum
Vertex is a minimum
Vertex is a minimum
-2
-3
-4
-5
-6
-7
-8
-9
-10
Vertex is a maximum
Remember Vertical &
Horizontal Shifts?
10
f (x)  x 3
g( x )   x  3
Y
9
 2
3
8
7
h( x )  x  4   3
3
6
5
4
3
2
1
X
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
Remember:
Stretching
Shrinking
??
f ( x )  x 2  4x  3
10
Y
9
f ( x )  x  2
8
7

2
7
6
5
4
3
2
1
X
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
f ( x )  x 2  12 x  3 3
-8
-9
-10
f ( x )   x  6 
2
3
The STD form of a Quadratic Equation:
f ( x )  a x  h
2
k
What is (h,k)?
Vertex:
(h,k)
Graph (without your battery operated brain for now…)
f (x)  2 x 2  8 x  7
YOUR
PAL the
AND
MINE!
Completing
Square
X-intercepts:
f ( x )  2 x  2 
Vertex :  2 , 1
2
1
?
10
Y
9
8
7
6
5
4
3
2
1
X
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9 10
Try another…
f ( x )  x 2  6 x  8
V = (3, 1) and x-intercepts are: (4, 0) and (2, 0)
Another:
f ( x )  2
Find an equation of a parabola that has
a V = (1, 2) and passes through the
origin.
x  12  2  2 x 2  4 x
Another:
Find an equation of a parabola that has
a V = (2, 3) and passes through (0, 2).
1
f ( x )   x  2
4

2
3
x2
 
 x 2
4
An easier way to find the vertex of a parabola:
f ( x )  ax 2  bx  c
b

V  
,f
 2a
b


 2a



f (x)  2 x 2  8 x  7
Vertex :
 2 ,
1
(x, y)
(x, f(x))
A baseball is hit at a point 3 feet above the ground at a
velocity of 100 fps at 45o. The path of the ball is
given by the following function:
f ( x )  0 .0 0 3 2 x 2  x  3
(Where f(x) is the height in ft, and x is the distance in
feet.) What is the maximum height of the ball? What is
the distance from the batter when the ball hits the ground?
What is the distance from the batter when the ball is at a
height of 3 feet?
Go!
Do!