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SFM Productions Presents:
Another joyous day continuing your Pre-Calculus experience!
2.1 Quadratic Functions and Models
Homework for section 2.1
p132 7-12 (all), 21-33 (eoo), 35-45, 49-55, 65, 71, 73, 76, 81, 84
The last three word problems are The last three word problems are important; I will check to make sure you important; I will check to make sure you did them - penalties will be assessed as did them - penalties will be assessed as needed.needed.
We are going to be looking at graphs of polynomial functions.
n n-1 1 0n-2 2
Let be a and let
a , a , a ,......a , a , a a
n nonne
ll be
gative integer
real numbers.
1 2 11 1 02( ) ...n n
n nf x a x a x a x a x a
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:
2( ) 3 5f x x x 2
( ) 3 5 4g x x
2
( ) 53x
h x 2( ) 8 17i x x
( ) 5 3j x x x
2( )f x ax bx c
I t's graph is a parabola.
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
2( ) , 0f x ax bx c a
2( )f x x
2( ) , 0f x ax bx c a
2( )f x x X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
Quadratic equations have an:axis of symmetry
X = 0
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
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10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
X = 0
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
X = -2
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
X = 6
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
Quadratic equations have:vertices, or in the singular: a vertex
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
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8
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10
0
Vertex is a minimumVertex is a minimumVertex is a minimum
Vertex is a maximum
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
3( )f x xRemember Vertical & Horizontal Shifts?
3( 2) 3xg x
3( 3) 4xh x
Remember:
Stretching
Shrinking ??
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
2( ) 4 3f x x x
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
2( ) 2 7f x x
2( ) 12 33f x x x
2
( ) 6 3f x x
The STD form of a Quadratic Equation:
2( )f x a x h k What is (h,k)?Vertex: (h,k)
Graph (without your battery operated brain for now…)
2( ) 2 8 7f x x x YOUR PAL AND MINE!Completing the Square
2( ) 2 12f x x
Vertex: 2 , 1
X-intercepts: ?
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
Try another… 2( ) 6 8f x x x
V = (3, 1) and x-intercepts are: (4, 0) and (2, 0)
Another:
Find an equation of a parabola that has a V = (1, 2) and passes through the origin.
2( ) 2 1 2f x x 22 4x x
Another:
Find an equation of a parabola that has a V = (2, 3) and passes through (0, 2).
21( ) 2 3
4f x x
2
24x
x
An easier way to find the vertex of a parabola:
V ,
2 2b b
fa a
(x, y)
(x, f(x))
2( ) 2 8 7f x x x
Vertex: 2 , 1
2( )f x ax bx c
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:
2( ) 0 .0032 3f x x x
(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!
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