Transcript Chapter #5

Chapter 5
Quadratic Equations and Functions
In This Chapter You Will …
• Learn to use quadratic functions to model
real-world data.
• Learn to graph and to solve quadratic
equations.
• Learn to graph complex numbers and to use
them in solving quadratic equations.
5.1 Modeling Data With
Quadratic Functions
What you’ll learn …
• To identify
quadratic functions
and graphs
• To model data with
quadratic functions
Quadratic Functions and
their Graphs
• A quadratic function is a function that can
be written in the standard form, where a≠0.
f(x) =
2
ax
Quadratic term
+ bx + c
Linear term
Constant term
Example 1 Classifying
Functions
Determine whether each function is linear or quadratic.
Identify the quadratic, linear, and constant terms.
y = (2x +3)(x – 4)
f(x) = 3(x2-2x) – 3(x2 – 2)
The graph of a quadratic function is a
parabola.
The axis of symmetry is
the line that divides a parabola into two
parts that are mirror images.
The vertex of a parabola is the point at
which the parabola intersects the axis of
symmetry.
The y-value of the vertex represents the
maximum or minimum value of the
function.
Example 2a
Graph y = 2x2 – 8x + 8
Vertex ___________
Axis of Symmetry ______
x
0
1
2
3
4
y
Example 3a Finding a
Quadratic Model
• Find a quadratic
function to model the
values in the table.
• Substitute the values
of x and y into
y = ax2 + bx + c.
• The result is a system
of three linear
equations.
X
2
3
4
Y
3
13
29
• 3 = a(2)2 + b(2) + c.
• 13 = a(3)2 + b(3) + c.
• 29 = a(4)2 + b(4) + c.
Answer: (3,-5,1) or
y = 3x2 - 5x + 1
5.2 Properties of Parabolas
What you’ll learn …
• To graph quadratic
functions
• To find maximum
and minimum values
of quadratic
functions
2.02 Use quadratic
functions and inequalities
to model and solve
problems; justify results.
• Solve using tables, graphs,
and algebraic properties.
• Interpret the constants
and coefficients in the
context of the problem.
Graphing Parabolas
• The standard form of a quadratic function is
y=ax2 + bx + c. When b=0, the function
simplifies to y=ax2 + c.
• The graph of y=ax2 + c is a parabola with an
axis of symmetry x =0, the y-axis. The
vertex of the graph is the y-intercept (0,c).
Properties Graph of a Quadratic
Function in Standard Form
The graph of y=ax2 + bx + c is a parabola when a≠0.
• When a>0, the parabola opens up.
• When a<0, the parabola opens down.
positive quadratic y = x2
negative quadratic y = –x2
Properties Graph of a Quadratic
Function in Standard Form
The graph of y=ax2 + bx + c is a parabola
when a≠0.
• The axis of symmetry is x= - b
2a
Properties Graph of a Quadratic
Function in Standard Form
The graph of y=ax2 + bx + c is a parabola
when a≠0.
• The vertex is ( - b2a , f(- b2a ) ).
Properties Graph of a Quadratic
Function in Standard Form
The graph of y=ax2 + bx + c is a parabola
when a≠0.
• The y intercept is (0,c).
Quadratic Graphs
y=x
The graph of a
quadratic function is a
U-shaped curve called
a parabola.
.
2
Example 1 Graphing a Function
of the Form y=ax2 + c
Graph y= -½x2 + 2
Graph y= 2x2 - 4
Symmetry
2
You can fold a parabola
so that the two sides
match evenly. This
property is called
symmetry. The fold or
line that divides the
parabola into two
matching halves is called
the axis of symmetry.
y=x +3
Vertex
The highest or lowest point of a parabola is its vertex,
which is on the axis of symmetry.
y=½x
Minimum
2
2
y = -4 x +3
Maximum
Determining Vertex and Axis of
Symmetry
Equation
y = -x2 + 4x + 2
2
y = -1/3x - 2x-3
y = 2x 2+ 8x -1
2
y = x - 2x - 3
Max/Min
Vertex
Axis of
YSymmetry Intercept(s)
5.3 Translating Parabolas
What you’ll learn …
• To use the vertex
form of a quadratic
function
2.02 Use quadratic
functions and inequalities
to model and solve
problems; justify results.
• Solve using tables, graphs,
and algebraic properties.
• Interpret the constants
and coefficients in the
context of the problem.
Investigation:
Vertex Form
Standard Form
y = ax2 +bx + c
b
2a
Vertex Form
y = a(x – h)2 + k
y = x2 -4x + 4
y = (x – 2)2
y = x2 +6x + 8
y = (x +3)2 - 1
y = -3x2 -12x - 8
y = -3(x +2)2 +4
y = 2x2 +12x +19
y = 2(x +3)2 +1
h
In other words …
To translate the graph of a quadratic
function, you can use the vertex form of a
quadratic function.
Properties
The graph of y = a(x – h)2 + k is the graph of
y = ax2 translated h units horizontally and k units
vertically.
• When h is positive the graph shifts right; when h is
negative the graph shifts left.
• When k is positive the graph shifts up; when the k
is negative the graph shifts down.
• The vertex is (h,k) and the axis of symmetry is the
line x=h.
Example 1a Using Vertex
Form to Graph a Parabola
Graph y = - 1 (x-2)2 +3
2
1. Graph the vertex.
2. Draw the axis of
symmetry.
3. Find another point.
When x=0.
4. Sketch the curve.
Example 1b Using Vertex
Form to Graph a Parabola
Graph y = 2 (x+1)2 - 4
1. Graph the vertex.
2. Draw the axis of
symmetry.
3. Find another point.
When x=0.
4. Sketch the curve.
Example 2a Writing the
Equation of a Parabola
• Write the equation
of the parabola.
• Use the vertex form.
• Substitute h=__ and
k= ___.
• Substitute x=0 and
y = 6.
• Solve for a.
Example 2b Writing the
Equation of a Parabola
• Write the equation
of the parabola.
• Use the vertex form.
• Substitute h=__ and
k= ___.
• Substitute x=___
and y = ___.
• Solve for a.
Example 2c Writing the
Equation of a Parabola
• Write the equation of a parabola that has
vertex (-2, 1) and goes thru the point (1,28).
• Write the equation of a parabola that has
vertex (-1, -4) and has a y intercept of 3.
Convert to Vertex Form
y = 2x2 +10x +7
y = -3x2 +12x +5
Convert to Standard Form
y = (x+3)2 - 1
y = -3(x -2 )2 +4
Example 3 Real World
Connection
• The photo shows the Verrazano-Narrows Bridge in New
York, which has the longest span of any suspension bridge
in the US. A suspension cable of the bridge forma a curve
that resembles a parabola. The curve can be modeled with
the function y = 0.0001432(x-2130)2 where x and y are
measured in feet. The origin of the function’s graph is at
the base of one of the two towers that support the cable.
How far apart are the towers? How high are they?
• Start by drawing a diagram.
• The function, y = 0.0001432(x-2130)2 , is in
vertex form. Since h =2130 and k =0, the vertex is
(2130,0). The vertex is halfway between the
towers, so the distance between the towers is
2(2130) ft = 4260 ft.
• To find the tower’s height, find y for x=0.
5.4 Factoring Quadratic
Expressions
• What you’ll learn …
• To find common and
binomial factors of
quadratic
expressions
• To factor special
quadratic
expressions
1.03 Operate with
algebraic expressions
(polynomial, rational,
complex fractions) to
solve problems.
Investigation: Factoring
1. Since 6 * 3 = 18, 6 and 3 up a factor pair for 18.
a. Find the other factor pairs for 18, including negative
integers.
b. Find the sum of the integers in each factor pair for 18.
2. Does 12 have a factor pair with a sum of -8? A sum of 9?
a. Using all the factor pairs of 12, how many sums are
possible?
b. How many sums are possible for the factor pairs of -12?
• Factoring is rewriting an expression as the
product of its factors.
• The greatest common factor (GCF) of an
expression is the common factor with the
greatest coefficient and the greatest
exponent.
Example 1a Finding Common Factors
3
2
4x + 12 x - 8
GCF ________
3
2
4b -2b -6b
GCF ________
Example 1b Finding Common Factors
3
3
2
3x - 12x +15x
6m - 12m - 24m
GCF
GCF
(
2
)
(
)
Example 2 Factoring when ac>0 and b>0
Factor x2 +8x +7
Factor x2 +6x +8
Factor x2 +12x +32
Factor x2 +14x +40
Example 3 Factoring when ac>0 and b<0
Factor x2 -17x +72
Factor x2 -7x +12
Factor x2 -6x +8
Factor x2 -11x +24
Example 4 Factoring when ac<0
Factor x2 - x - 12
Factor x2 -14x - 32
Factor x2 +3x - 10
Factor x2 +4x - 5
Example 5 Factoring when a≠0 and ac>0
Factor 2x2 +11x + 12
Factor 4x2 +7x + 3
Factor 3x2 - 16x +5
Factor 2x2 - 7x + 6
Example 6 Factoring when a≠0 and ac<0
Factor 4x2 -4x - 15
Factor 3x2 - 16x - 12
Factor 2x2 +7x - 9
Factor 4x2 +5x - 6
Special Cases
• A perfect square trinomial is the product
you obtain when you square a binomial.
• An expression of the form a2 - b2 is defined
as the difference of two squares.
Factoring a Perfect Square
Trinomial with a = 1
2
x - 8x + 16
n 2 - 16n + 64
The Difference of Two
Squares
2
x - 121
(
)(
2
4x - 36
)
(
)(
)
5.5 Quadratic Equations
What you’ll learn …
• To solve quadratic
equations by factoring
and by finding square
roots
• To solve quadratic
equations by graphing
2.02 Use quadratic
functions and inequalities
to model and solve
problems; justify results.
• Solve using tables, graphs,
and algebraic properties.
• Interpret the constants
and coefficients in the
context of the problem.
• The standard form of a quadratic equation is
ax2 + bx + c = 0, where a ≠ 0. You can solve some
quadratic equations in standard form by factoring
the quadratic expression and then using the ZeroProduct Property.
Zero-Product Property
If ab = 0, then a =0 or b=0.
Example
If (x +3) (x -7) = 0 then (x +3) = 0 or (x -7) = 0.
Zero Product Property
• ( x + 3)(x + 2) = 0
• (x + 5)(2x – 3 ) = 0
Example 1a Solve by Factoring
2
x – 8x – 48 = 0
2
x + x – 12 = 0
Example 1b Solve by Factoring
2x – 5x = 88
2
2
x - 12x = -36
Example 2 Solving by Finding
Square Roots
2
x – 25 = 0
2
5x - 180 = 0
2
x +4=0
Example 4 Solve by Graphing
x 2– 4 = 0
x2 = 0
x2 + 4 = 0
The number of x intercepts determines the number of solutions!!
Using the Calculator
Solve:
• 1. Set y=
and graph
with a standard window.
• 2. Use the ZERO command to find
the roots -- 2nd TRACE (CALC), #2
zero
• 3. Left bound? Move the spider as
close to the root (where the graph
crosses the x-axis) as possible. Hit
the left arrow to move to the "left" of
the root. Hit ENTER. A "marker" ►
will be set to the left of the root.
4. Right bound? Move the spider as close to the
root (where the graph crosses the x-axis) as
possible. Hit the right arrow to move to the
"right" of the root. Hit ENTER. A "marker"
◄ will be set to the right of the root.
5. Guess? Just hit ENTER.
6. Repeat the entire process to find the second
root (which in this case happens to be x = 7).
Using a Graphing Calculator
Solve Each Equation
x2 + 6x + 4 = 0
3x2 + 5x - 12 = 8
5.6 Complex Numbers
What you’ll learn …
• To identify and graph
complex numbers
• To add, subtract, and
multiply complex
numbers
1.02 Define and
compute with complex
numbers.
• When you learned to count, you used
natural numbers 1,2,3, and so on. Your
number system has grown to include other
types of numbers. You have used real
numbers, which include both rational
numbers such as ½ and irrational numbers
such as √2.
• Now your number system will expand to
include numbers such as √-2.
• The imaginary number i is defined as the
number whose square is -1. So i2 = -1and
i = √-1. An imaginary number is any
number of the form a + bi where b≠0.
• Imaginary numbers and real numbers
together make up the set of complex
numbers.
Example 1 Simplifying
Numbers Using i
√-9
√-18
√-12
√-2
Example 2 Simplifying
Imaginary Numbers
√-9 + 6
√-18 + 7
• The diagram below shows the sets of
numbers that are part of the complex
number system and examples of each set.
• You can use the complex
number plane to represent a
complex number geometrically.
• Locate the real part of the
number on the horizontal axis
and the imaginary part on the
vertical axis.
• You graph 3 – 4i in the same
way you would graph (3,-4) on
the coordinate plane.
• The absolute value of a
complex number is its
distance from the origin on
the complex number plane.
• You can find the absolute
value by using the
Pythagorean Theorem.
• In general, a +bi = a2+b2
Example 3 Finding Absolute Values
Find 5i
Find 3i - 4
Find -2 + 5i
Example 4 Additive Inverse of a
Complex Number
Find the additive inverse of -2 +5i.
Find the additive inverse of 4 – 3i.
Find the additive inverse of a + bi.
Example 5 Adding Complex Numbers
Simplify (5 + 7i) + (-2 + 6i)
Simplify (8 + 3i) - (2 + 4i)
Simplify 7 - (3 + 2i)
Example 6 Multiplying Complex Numbers
Find (5i) + (-4i)
Find (2 + 3i) - (-3 + 5i)
Find (6 – 5i) (4 – 3i)
Example 7 Finding Complex Solutions
Solve 4x2 + 100 = 0
Solve 3x2 + 48 = 0
Solve -5x2 - 150 = 0
5.7 Completing the Square
What you’ll learn …
• To solve equations by
completing the square
• To rewrite functions
by completing the
square
2.02 Use quadratic
functions and inequalities
to model and solve
problems; justify results.
• Solve using tables, graphs,
and algebraic properties.
• Interpret the constants
and coefficients in the
context of the problem.
Perfect Square Trinomials
Examples
 x2 + 6x + 9
 x2 - 10x + 25
 x2 + 12x + 36
Creating a Perfect Square Trinomial
In the following perfect square
trinomial, the constant term is missing.
X2 + 14x + ____
 Find the constant term by squaring half
the coefficient of the linear term.
 (14/2)2
X2 + 14x + 49

Perfect Square Trinomials
Create perfect square trinomials.
 x2 + 20x + ___
 x2 - 4x + ___
 x2 + 5x + ___
Example 1 Solving a Perfect Square
Trinomial Equation
Step 1: Factor the trinomial.
Step 2: Find the Square Root
of each side.
Step 3: Solve for x.
2
2
+
Example 2a Completing the Square
2
b
• Find
.
2
Substitute -8 for b.
• Complete the square.
Example 2b Completing the Square
2
b
• Find
.
2
Substitute for b.
• Complete the square.
Example 3 Solving by Completing
the Square
Solve the following equation by
completing the square:
Step 1: Rewrite so all terms
containing x are on one side.
x  8 x  20  0
2
x  8 x  20
2
Example 3 Continued
Step 2: Find the term
that completes the
square on the left side
of the equation. Add
that term to both sides.
x  8x 
2
=20 +
1
 (8)  4 then square it, 42  16
2
x  8 x  16  20  16
2
Example 3 Continued
Step 3: Factor the
2
perfect square trinomial x  8 x  16  20  16
on the left side of the
(
x

4)(
x

4)

36
equation. Simplify the
2
right side of the
( x  4)  36
equation.
Step 4: Take the square
root of each side.
( x  4)  36
2
( x  4)  6
Example 3 Continued
Step 5: Solve for x.
x  4  6
x  4  6 and x  4  6
x  10 and x=2
Solve each by Completing the Square
x2 + 4x – 4 = 0
x2 – 2x – 1 = 0
Example 4 Finding Complex Solutions
x2 - 8x + 36 = 0
x2 +6x = - 34
Example 5 Solving When a≠0
5x2 = 6x + 8
2x2 + x = 6
• In lesson 5-3 you converted quadratic
b
functions into vertex form by using x = 2a
to find the x-coordinate of the parabola’s
vertex.
• Then by substituting for x, you found the y
coordinate of the vertex.
• Another way of rewriting a function is to
complete the square.
Example 6a Rewriting in Vertex Form
• Complete the square.
• Add and subtract 3
on the right side.
• Factor the perfect
square trinomial.
• Simplify.
2
x2 + 6x + 2
Example 6b Rewriting in Vertex Form
y = x2 - 10x - 2
y = x2 + 5x + 3
5.8 The Quadratic Formula
What you’ll learn …
• To solve quadratic
equations by using the
quadratic formula
• To determine types of
solutions by using the
discriminant
2.02 Use quadratic
functions and inequalities
to model and solve
problems; justify results.
• Solve using tables, graphs,
and algebraic properties.
• Interpret the constants
and coefficients in the
context of the problem.
The Quadratic Formula
Example 1a Using the Quadratic
Formula
x 2 – 2x – 8 = 0
-b ± √ (b) – 4 (a) (c)
2(a)
2
-( )±
√
(
2
) –4(
2( )
)(
)
Example 1b Using the
Quadratic Formula
2
x – 4x – 117 = 0
2
(
)
±
√
(
)
–4(
2
-b ± √ (b) – 4 (a) (c)
( )
2(a)
2( )
)
Example 2a Finding Complex Solutions
2x2 = -6x - 7
-b ± √ (b) – 4 (a) (c)
2(a)
2
Example 2b Finding Complex Solutions
-2x2 = 4x + 3
-b ± √ (b) – 4 (a) (c)
2(a)
2
• Quadratic equations can have real or
complex solutions. You can determine the
type and number of solutions by finding the
discriminant.
-b
+
√
x=
2
b
2a
– 4ac
◄ the discriminant
Value of the
Discriminant
Type and Number of
Solutions for ax2 + bx + c
b2 – 4ac > 0
Two real solutions
b2 – 4ac = 0
One real solution
b2 – 4ac < 0
No real solution;
Two imaginary
solutions
Examples of Graphs of Related
Functions y=ax2 + bx + c
Example 4 Using the Discriminant
x 2 +6x + 8 = 0
x 2 +6x + 10 = 0
Methods for Solving Quadratics
Discriminant
Methods
Positive square number
Factoring, Graphing, Quadratic Formula, or
Completing the Square
Positive non-square
number
For approximate solutions: Graphing, Quadratic
Formula, or Completing the Square
For exact solutions: Quadratic Formula, or
Completing the Square
Zero
Factoring, Graphing, Quadratic Formula, or
Completing the Square
Negative
Quadratic Formula, or Completing the Square
In This Chapter You
Should Have …
• Learned to use quadratic functions to model
real-world data.
• Learned to graph and to solve quadratic
equations.
• Learned to graph complex numbers and to
use them in solving quadratic equations.