Transcript Chapter 5

Chapter 5
Quadratic Equations and
Functions
5-1 Warm Up
What is a quadratic equation? What
does the graph look like? Give a real
world example of where it is applied.
5-1 Modeling Data with Quadratic
Functions
OBJ: Recognize and use quadratic
functions
Decide whether to use a linear or
a quadratic model
Quadratic Functions
Quadratic function is a function that
can be written in the form f(x)=
ax²+bx+c, where a ≠ 0
The graph is a parabola
The ax² is the quadratic term
The bx is the linear term
The c is the constant term
The highest power in a quadratic
function is two
A function is linear if the greatest
power is one
Tell whether each function is linear
or quadratic
F(x)= (-x+3)(x-2)
Y=(2x+3)(x-4)
F(x)=(x²+5x)-x²
Y= x(x+3)
Modeling Data
Last semester you modeled data that,
when looking at the scatter plot, the
data seemed to be linear
Some data can be modeled better
with a quadratic function
Find a quadratic model that fits the
weekly sales for the Flubbo Toy
Comp
Week
1
2
3
4
5
6
7
8
Sales (in millions)
$15
$24
$29
$31
$30
$25
$16
$5
5-1 Wrap Up
What is a quadratic function?
What kinds of situations can a
quadratic function model?
5-2 Warm Up
List as many things as you can that
have the shape of a parabola
5-2 Properties of Parabolas
OBJ: Find the min and max value of a
quadratic function
Graph a parabola in vertex form
Comparing Parabolas
Any object that is tossed or thrown will
follow a parabolic path.
The highest or lowest point in a parabola is
the vertex
It is the vertex that is the maximum or
minimum value
If a is positive the parabola opens up,
making the vertex a min point
If a is negative the parabola opens down,
making the vertex a max point
Axis of symmetry divides a parabola
into two parts that are mirror images
of each other
The equation of the axis of symmetry
is x=(what ever the x coordinate of
the vertex is)
Two corresponding points are the
same distance from the axis of
symmetry
Y=ax²+bx+c is the general equation of
a parabola
If a>0 the parabola opens up
If a<0 the parabola opens down
If a is a fraction it is a wide opening
If a is a whole number it is narrow
Examples
Each tower of the Verrazano Narrows
Bridge rises about 650 ft above the
center of the roadbed. The length of
the main span is 4260 ft. Find the
equation of the parabola that could
model its main cables. Assume that
the vertex of the parabola is at the
origin.
Translating Parabolas
Not every parabola has its vertex at
the origin
Y=a(x-h)²+k is the vertex form of a
parabola
It is a translation of y=ax²
(h,k) are the coordinates of the vertex
the image of the parabola y  ax under
the Translatio n Th,k is the parabola with
2
equation
y - k  a(x - h)
2
Sketch the following graphs
y  7  3( x  6)
2
y  2  2( x  3)
2
y  k  a( x  h) 2 is the vertex form of an equation
for a parabola.
The vertex is ( h, k)
Axis of symmetry is x = h
If a > 0 it is a max
If a < 0 it is a min
Graph, give equation for axis of
symmetry and state the vertex.
1 2
y3 x
2
1
2
y   x  3
2
Example
Sketch the graph of y= -1/2(x-2)²+3
Sketch the graph of y = 3(x+1)²-4
Wrap Up 5-2
What does the vertex form of a
quadratic function tell you about its
graph?
Warm Up 5-3
List formulas that you know to use to
find answers to problems quickly. (list
as many formulas as you can)
5-3 Comparing Vertex and
Standard Forms
OBJ: Find the vertex of a function
written in standard form
Write equations in vertex and
standard form
Get into a group of four
Turn to page 211
Do part a
What do you notice about the graphs of
each pair of equations?
What is true of each pair of equations?
Write a formula for the relationship
between b and h
How can we modify our formula to show
the relationship among a, b, and h. (the last
couple of equations)
Standard form of a parabola
When a parabola is written as
y=ax²+bx+c it is standard form
The x coordinate of the vertex can be
found by –b/(2a)
To find the y coordinate by – [(b^24ac)/4a]
Suppose a toy rocket is
launched to its height in
meters after t seconds is given
by H = -4.9t^2 +20t +1.5. How
high is the rocket after one
second? How high is the
rocket when launched. How
high is the rocket after 12
seconds?
Example
Write the function y= 2x²+10x+7 in
vertex form
Write the function y= -x²+3x-4 in
vertex form
What is the relationship between the
axis of symmetry and the vertex of the
parabola?
Example
As a graduation gift for a friend, you plan to
frame a collage of pictures. You have a 9 ft
strip of wood for the frame. What
dimensions of the frame give you
maximum area of the collage?
What is the maximum area for the collage?
What is the best name for the geometric
shape that gives the maximum area for the
frame?
Will this shape always give the max area?
Consider this general formula:
1 2
h   gt  v0t  h0
2
A ball is dropped form the top of a
20 meter tall building. Find an
equation describing the relation
between the height and time.
Graph its height h after t
seconds. Estimate how much
time it takes the ball to fall to the
ground. Explain your reasoning
Write y= 3(x-1)²+12 in standard form
A rancher is constructing a cattle pen
by a river. She has a total of 150 ft of
fence, and plan to build the pen in the
shape of a rectangle. Since the river
is very deep, she need only fence
three sides of the pen. Find the
dimensions of the pens so that it
encloses the max area.
Show that y  2(x  3) and
2
y  2x  12x  10 are equivalent
2
Suppose a swimming pool
50 m by 20 m is to be built
with a walkway around it. IF
the walkway is w meters
wide, write the total area of
the pool and walkway in
standard form
Consider this
If a quarterback tosses a football to a
receiver 40 yards downfield, then
the ball reaches a maximum height
halfway between the passer and
the receiver, it will have a equation
h  0.025 x  x  6
2
Example
Suppose a defender is 3
yards in front of the
receiver. This means the
defender is 37 yards from
the quarterback. Will he be
able deflect or catch the
ball?
Examples
A model rocket is shot at an
angle into the air from the
launch pad. The height of the
rocket when it has traveled
horizontally x feet from the
launch pad is given by
2
h  .163x  11.43x
Graph this equation
A 75-foot tree, 10 feet from
the launch pad is in the path
of the rocket. Will the rocket
clear the top of the tree?
Estimate the maximum height
the that the rocket will
reach.
Wrap Up 5-3
Describe the similarities and
differences between the vertex form
and standard form of quadric
equations.
Warm Up 5-4
Name mathematical operations that
are opposites of each other. For
example, addition is the opposite of
subtraction.
Two inverse functions are opposite of
each other in the same way.
5-4 Inverses and Square Root
Functions
OBJ: Find the inverse of a function
Use square root functions
Consider the functions
F(x)= 2x-8
G(x)= (x+8)/2
Find F(6) and G(4)
F(x) and G(x) are inverses because one
function undoes the other
Graph each function on the same
coordinate plane
Find three coordinates on f(x)
Reverse the coordinates and graph
What do you notice?
Definition
The inverse of a relations
is the relation obtained
by reversing the order of
the coordinates of each
ordered pair in the
relation
Remember
If the graph of a function contains a
point (a,b), then the inverse of a
function contains the point (b,a)
Example
Let f  1,4, 2,8, 3,8, 0,0,  1,4
Find the inverse of f

Inverse Relation Theorem
Suppose f is a relation and g
is the inverse of f. Then:
1. A rule for g can be found by
switching x and y
2. The graph of g is the
reflection image of the
graph of f over y=x
3. The domain of g is the
range of f, and the range of
g is the domain of f
Remember
The inverse of a relation
is always a relation
The inverse of a function
is not always a function
Examples
Consider the function with
equation y= 4x-1. Find an
equation for its inverse.
Graph the function and its
inverse on the same
coordinate plane. Is the
inverse a function?
Consider the function with
domain the set of all real
numbers and equation
y=x^2
What is the equation for the
inverse? Graph the function
and its inverse on the same
coordinate plane. Is the
inverse a function? Why or
Why not?
Example
Graph the function and its inverse.
The write the equation of the inverse
x4
y
2
More Examples
Find the inverses of these functions
f ( x)  x  3
2
y  3x  2
2
Square Root Functions
Y=x is the square root function
The graph starts at (0,0)
The domain is x0
The range is y0
Example
Graph the function and state the
domain and range
y  x4 2
5-4 Wrap Up
What can you tell me about a function
and its inverse?
Warm Up 5-5
Brain storm all the methods you know
for solving this equation. Include less
efficient methods. We will vote on
which you all prefer.
x  6x  8  0
2
5-5 Quadratic Equations
OBJ: Solve quadratic equations by
factoring, finding square roots, and
graphing
Zero Product Property
For all real numbers a and b. If ab=0,
then a=0 or b=0
Example
(x+3)(x-7)=0
(x+3)=0 or (x-7)=0
We can use the Zero Product
Property to solve quadratic equations.
That is why we learned how to factor.
Important rule for factor
quadratics
Make sure the quadratic is = 0
if not add or subtract until all numbers
and variables are on the same side.
Examples:
Solve each quadratic by factoring
x x6
2
x  6 x  8
2
Solve
x  14 x  33  0
2
r  11r  12  0
2
10t  80t  200t  0
3
2
Solving Quadratics by square roots
When equations are in the form of y =
ax² you can just divide by a, then take
the square root.
You will have two answers
Solving Quadratic
Equations
Solve :
3x  18
2
x  40
2
Example
Smoke jumpers are firefighters who
parachute into areas near forest fires.
Jumpers are in free fall from the time
they jump from a plane until they
open their parachutes. The function
y= -16x²+1600 gives jumper’s height
y in feet after x seconds for a jump
from 1600 ft. How long is the free fall
if the parachute opens at 1000 ft?
Another way to solve
Graphing is another way to solve
quadratics.
The solutions would be at the x
intercepts of the parabola
Solve by graphing
The last example and
x2 x

x
3
5-5 Wrap Up
Describe how the zero product
property can be used to solve
quadratic equations and which
method of solving quadratics do you
prefer? Why?
Warm Up 5-6
Think about the square root of a
negative number
How do you think you could write the
square root of a negative number?
What would the square root mean?
Please be creative with your
responses
5-6 Complex Numbers
OBJ: Identify and graph complex
numbers
Add, subtract, and multiply
complex numbers
Identifying Complex Numbers
The system you use now is called the
real number system
Real number system is the rational,
irrational, integers, whole, and natural
numbers
We will now expand our knowledge to
include numbers like √-2
Examples
Solve :
t  400
2
2 x  100  0
2
Show that i 3 is a square root of - 3
Simplify
8i 
2
 3  49
 9   121
9   121
- 144
16
The imaginary number i is defined as
the principal square root of -1
i=√-1, and i²=-1
Other imaginary numbers include -5i,
i√2, and 2+3i
Numbers in the form a+bi form are
called Complex Numbers
A+Bi (Complex Numbers)
All real numbers are complex
numbers where b=0
5+0i=5
An imaginary number is also in the
form a+bi, but b≠0
0+5i=5i
Complex Numbers
Real Numbers
Imaginary Numbers
Rational Numbers
Integers
Whole Numbers
Natural
Simplify each number
4
-8
- 25
Graphing Complex Numbers
They are graphed like regular points
The x coordinate is the real number
part
The y coordinate is the imaginary
number part
3+5i would have the point (3,5)
Recall
The absolute value of a real number
is its distance from zero on a number
line
The absolute value of a complex
number is its distance from the origin
on the complex number plane
Formula to find distance
a  bi  a  b
2
2
Find
|5i|
|3-4i|
|-3i|
|8+6i|
Operations with Complex Numbers
To add or subtract complex numbers,
combine the real parts and the
imaginary parts separately
Simplify the following expressions
(5+7i)+(-2+6i)
(8+3i)-(2+4i)
Operations with
Complex Numbers
3 +4i) + (7 + 8i)
2i(8 + 5i)
(6-5i)+(3 +4i)
(5+9i)(2-7i)
(1+i)(1-i)
(
Multiply (5i)(-4i)
Multiply (2+3i)(-3+5i)
Simplify (3-2i)(-2+4i)
(6-5i)(4-3i)
(4-9i)(4+9i)
Solving quadratic equations using
complex numbers
Solve 4k²+100=0
3t²+48=0
5x²=-150
Wrap Up 5-6
Describe the parts of a complex
number and explain what they
represent.
Warm Up 5-7
(x+7)²
(x+7)(x+7)
x²+14x+49
How can you determine that this is
equivalent to (x+7)²
5-7 Completing the Square
OBJ: Solving quadratic equations by
completing the square
Rewriting quadratic equations in
vertex form
Completing the Square
To complete the square on x  bx,
1 2
add ( b)
2
1 2
1 2
2
x  bx  ( b)  ( x  b)
2
2
2
Rewrite the following equations
and state the vertex.
y  x  18 x  90
2
y  x  11x  4
2
y  3 x  12 x  1
2
y  5 x  4 x  3
2
Solve the following:
x²=8x-36
x²-4x=-8
5x²=6x+8
A local florist is deciding how much
money to spend on advertising. The
function p(d)=2000 +400d-2d² models
the profit that the store will makes as
a function of the amount of money it
spends. How much should the store
spend on advertising to maximize its
profit?
Real World Example
Suppose a ball is thrown
straight up form a height
of 4 feet with an initial
velocity of 50 feet per
second. What is the
maximum height of the
ball?
Wrap Up 5-7
Explain how to solve quadratic
equation by completing the square.
Warm Up 5-8
Given 4x²+2x+3=0
What are the values of a, b, and c
Find –b, b², 4ac,
b²-4ac
√ b²-4ac
2a
-b+ √ b²-4ac, -b- √ b²-4ac
5-8 The quadratic formula
OBJ: Solving quadratic equations using
the quadratic formula
Determine types of solutions using
the discriminant.
What is a Quadratic Equation
A quadratic equation is an
equation that can be
written in the form
ax  bx  c  0
2
Quadratic Formula
You can use the quadratic formula to
calculate x using a, b, and c. YOU
SHOULD MEMORIZE THIS FORMULA
If ax  bx  c  0 & a  0
2

 b  b  4ac
x
2a
2

Solve
 x  2 x  27  0
2
3 x  6 x  45  0
2
2r  11r  12  0
2
m  3m  14
2
5k  5k  2k  2k
2
2
Solve
2x²= -6x -7
2x²+4x=-3
Solve 10x^2-13x-3 =0
Accounting for a driver’s reaction
time, the minimal distance in feet
it takes for a certain car to stop
is approximated by the formula
d=.042s^2 + 1.1s+4, where s is
the speed in miles per hour. If a
car took 200 feet to stop, about
how fast was it traveling?``
Discriminant Property
If ax  bx  c  0, Then
2
When b  4ac  0
2
When b  4ac  0
2
When b  4ac  0
2
Has 2 real
solutions
Has one real
solution
Has 2 complex
solution
Without solving determine how
many real solutions the
equations have
8x  5x  2  0
2
25 x  10  1  0
2
The amount of power watts generated
by a certain electric motor is molded
by the equation P(l)=120l-5l² where l
is the amount of current passing
through the motor in amperes (A).
How much current should you apply
to the motor to produce 600 W of
power?
A scoop is a field hockey pass that propels
the ball from the ground into the air.
Suppose a player makes a scoop that
releases the ball with an upward velocity of
34 ft/sec. The function h = -16t²+34t
models the height h in feet of the ball at
time t in seconds. Will the ball ever reach a
height of 20ft? If so how many seconds will
it take? Will it reach 15 ft? How long will it
take?
Lets use the graphing calculators
x²+6x+8=0
x²+6x+9=0
x²+6x+10=0
A Way to Sum it Up
Discriminant
Methods of Solving
Positive square #
Factoring, graphing, quad formula,
complete the square
Positive non square #
Graphing, quad formula, or
completing the square (all will yield
an approx solution)
Quad formula or completing the
square (will both yield an exact
solution)
Zero
Factoring, graphing, completing the
square, or quad formula
Negative
Quad formula or completing the
square
Wrap Up 5-8
Describe how to use the quadratic
formula to solve quadratic equations.