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Transcript
```Notes Section 5.1: Modeling Data with Quadratic Functions
Example One: Find a quadratic model for the following:
a.
x
-1 0 2
f(x) 1 -1 7
(-1, 1)
(0, -1)
(2 , 7)
1= a(-1)2 + b(-1) + c
-1 = a(0)2 + b(0) + c
7 = a(2)2 + b(2) + c
or
or
or
1=a–b+c
-1 = c
7 = 4a + 2b + c
Finish:
b. (-1, 10) (2, 4) (3, -6)
Example Two: Identify the vertex and axis of symmetry of each parabola:
Vertex ( _____ , ______)
Axis of symmetry :________
Vertex ( _____ , ______)
Axis of symmetry :________
Vertex ( _____ , ______)
Axis of symmetry :________
Example Three: Determine whether the function is quadratic, linear. Identify the quadratic, linear, and constant terms.
a. y=(x-2)(x+4) b. y = 3x(x-7)
c.
D.
Example Four: A toy rocket is shot upward from ground level. The table shows the height of the rocket at different
times:
Time (seconds) 0 1
2
3
4
Height (feet)
0 256 480 672 832
a. Find a quadratic model for the data
b. Use the model to estimate the height of the rocket after 1.5 seconds
Height
Time
Complete the equation of each function using the given point. (Find c)
1)
2)
and (3, 10)
and (-2, -15)
3)
4)
and
and (4, 15)
Notes 5.2 Properties of Parabolas
General Form of a Parabola:
Opens Up:
and Opens Down:
Vertex:
Axis of Symmetry:
y-intercept:
Example One: Graph
Vertex:
y-intercept:
axis of symmetry:
Chart of Points:
Example Two: Graph
Vertex:
y-intercept:
axis of symmetry:
Chart of Points:
(0 , c )
Example Three: Graph
Vertex:
y-intercept:
axis of symmetry:
Chart of Points:
Example Four: The number of weekend get-away packages a hotel can sell is modeled by -0.12p + 60, where p is the
price of a get-away package. What price will maximize the revenue? What is the maximum revenue?
Revenue = Price
# of packages sold
Revenue =
Example Five: The number of Albert Pujols bobble head figures the St. Louis Cardinals can sell can be modeled by
-4p+100, where p is the price of the bobble head. What price will maximize revenue? What is the maximum revenue?
Notes 5.3: Transforming Parabolas
Vertex Form: _______________
Equation
Axis of Symmetry
Vertex
y-intercept
a<0
a>0
k
h
Examples:
Fill out the information, then graph the parabola.
1)
Vertex: _______
Axis of symmetry: _______
y-intercept: ________
point: (1,
x
y
) Its reflection: _______
2)
Vertex: _______
Axis of symmetry: _______
y-intercept: ________
Its reflection: ________
point: (1,
)
Its reflection: ________
x
y
Write the equation of each parabola in vertex form.
3)
4)
plug in h = 3 and k=4
plug in (1, -4)
Solve for “a”
The Equation is: y = _________________
Standard form: ___________________ Vertex form: ___________________
Change each equation from standard form to vertex form.
1)
2)
3
5)
6)
7)
8)
Change each equation into standard form.
9)
10)
Review:
F
O
I
L
1)
2)
3)
Factor:
Examples:
1)
2)
3)
4)
5)
6)
More Practice: Factor each expression.
1)
2) 2
3) 4
4) 3
5)
6) 4
7)
8)
9)
10)
by factoring, square roots, and graphing!!!!
Factoring: get the equation = 0, factor, (
)(
Square roots: rewrite as
, take square root of both sides, +/-
, isolate
)=0, set each individual =0
Graphing: 2nd calc intersect
Example One:
Solve by factoring
x = _____ or x = _____
Example Two:
Solve the following by factoring:
A.
B.
C.
Example Three: Solve by finding square roots.
Solve:
Example Four: The function
models the height of a heavy object t seconds after it is dropped from a
building that is 270 feet tall. How long until it hits the ground.
270 ft
t = ____ for a height of 0
t
Example Five: Use Square roots to solve the following:
A.
B.
Example Six: Solve by graphing
A carpenter wants to cut a piece of plywood in the shape of a right triangle as shown. He wants the hypotenuse of the
triangle to be 6 feet long. About how long should the perpendicular sides be?
X+1
6 ft
x
Example Seven: Solve by graphing
A.
B.
Y1 = __________ Y3 = __________
Y1 = __________ Y3 = __________
2nd calc: Intersect
2nd calc: Intersect
x = ____ or x = _____
x = ____ or x = _____
C.
Y1 = _________ Y3 = ____________ 2nd Calc Intersect
x= ____ or x = _____
Extra Practice (5.1-5.5)
Name ___________________
1) Write the equation of the quadratic function that contains the 3 given points (use elimination).
(1, 9)
(-1, 3)
(-2, 6)
2) Complete the equation of each function using the given point. (Find c)
, and (2, -1)
3) Fill in the information about the quadratic, and then graph the parabola.
y  2 x 2  6
axis: _______
vertex: ______
point: (1, )
reflection: _______
point: (2, )
reflection: _______
4) Fill in the information about the quadratic, and then graph the parabola.
y
1 2
x  2x  6
2
axis: _______
vertex: ______
y-int: _______ reflection: _______
point: (-2,
) reflection: _______
5) Fill in the information about the quadratic, and then graph the parabola.
y  ( x  2) 2  2
axis: ________ vertex: _______
y-int: _______ reflection: ________
point: (1,
)
reflection: ________
6) Change the equation to vertex form.
y  2 x 2  8x  3
7) Change the equation to standard form.
y  3( x  1) 2  2
Factor each expression completely.
8) x 2  5 x  24
10) Solve the quadratic by factoring.
x 2  9 x  18  0
9) 2 x 2  7 x  15
Notes 5.6 Complex Numbers:
Square root of a negative real number:
a+bi
=
Imaginary Number:
Complex Number:
Example One: Simplifying Numbers
a.
d.
b.
c.
e.
f.
Example Two: Simplifying Imaginary Numbers. (Rewrite as a + bi )
a.
b.
c.
Example Three: Operations with Complex Numbers
a.
b.
c.
d.
f.
g.
h.
i.
Example Four: Finding Complex Solutions NO b TERM!!!!!
a.
c.
+54=0
b.
d.
Solving
using the quadratic formula and our brains!!!!
To determine the Possible Number of Real Solutions Graphically-Look at the x- intercepts:
Two x-intercepts
2 Real Solutions
Zero x-intercepts
0 Real Solutions & 2 Complex
One x-intercept
1 Real Solution
To determine the Possible Number of Real Solutions look at the Discriminant:
discriminant
2 Real Solutions
BONUS INFORMATION:
0 Reals, 2 Complex
1 Real Solution
PERFECT SQUARE, then we can solve using factoring
Example: Solve using Factoring or the Quadratic Formula
3x2-x=4
-2x2=4x+3
4x2=8x-3
x2+4x=41
Example Two: Determine the type and number of solutions of each equation:
x2+6x+9
x2+6x+10=0
A player throws a ball up and toward a wall that is 17 feet high. The height h in feet of the ball t seconds after it leaves
the player’s hand is modeled by h = -16t2+25t+6. If the ball makes it to where the wall is, will it go over the wall or hit
the wall?
h = -16t2+25t+6
17= -16t2+25t+6
Example Three: The longer leg of a right triangle is 1 unit longer than the shorter leg. The hypotenuse is 3 units long.
What is the length of the shorter leg?
5.6-5.8 Review
1) Solve each equation by factoring or by taking square roots.
a)
b)
c)
d)
e)
f)
2) Simplify each expression.
a)
b)
c)
d)
e)
f)