Download ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
Goal  Graph quadratic functions.
VOCABULARY
Quadratic function
A function that can be written in the standard form y = ax2 + bx+ c where a  0
Parabola
The U-shaped graph of a quadratic function
Vertex
The lowest or highest point on a parabola
Axis of symmetry
The vertical line that divides the parabola into mirror images and passes through the vertex
Minimum and maximum value
For y= ax2 + bx+ c, the vertex's y-coordinate is the minimum value of the function if
a  0 and its maximum value if a  0.
PARENT FUNCTION FOR QUADRATIC FUNCTIONS
The parent function for the family of all quadratic functions is f(x) = __x2__ . The graph is shown below
The lowest or
highest point on a
parabola is the
vertex for
f(x) = x2 is (0,0)
The axis of
symmetry divides
the parabola into
mirror images and
passes through the
vertex
For f(x) = ax2, and for any quadratic function g(x)= ax2 + bx + c where b = 0, the vertex lies on the __y-axis__
and the axis of symmetry is x = _0_.
Example 1
Graph a function of the form y = ax2 + c
Graph y = 2x2 + 2. Compare the graph with the graph of y = x2.
PROPERTIES OF THE GRAPH OF y = ax2 + bx + c
Characteristics of the graph of y = ax2 + bx + c:
 The graph opens up if a __ 0 and opens down if a __ 0.
2
 The graph is narrower than the graph of y = x if | a | __ 1 and wider if | a | __ 1.
b
b
 The axis of symmetry is x = 
and the vertex has x-coordinate 
.
2a
2a
 The y-intercept is _c_. So, the point (0, _c_) is on the parabola.
Example 2
Graph a function of the form y = ax2 + bx + c
Graph y = x2 + 4x  3.
MINIMUM AND MAXIMUM VALUES
Words For y = ax2 + bx + c, the vertex's y-coordinate is the minimum value of the function if a __ 0 and the
maximum value if a __ 0.
Example 3
Find the minimum or maximum value
Tell whether the function y = 3x2 + 12x  6 has a minimum value or a maximum value. Then find the
minimum or maximum value.
Checkpoint Complete the following exercises.
Graph the function. Label the vertex and axis of symmetry
2.
1. y = x2
y = x2  4x + 2
3.
Find the minimum value of y = 2x2  6x + 6
4.2 Graph Quadratic Functions in Vertex or Intercept Form
Goal  Graph quadratic functions in vertex form or intercept form.
VOCABULARY
Vertex form
A quadratic function written in the form y = a(x  h)2 + k
Intercept form
A quadratic function written in the form y = a(x  p)(x  q)
GRAPH OF VERTEX FORM
y = a(x  h)2 + k
The graph of y = a(x  h)2 + k is the parabola y = ax2 translated _horizontally_ h units and _vertically_ k units.
 The vertex is (_h_, _k_ ).
 The axis of symmetry is x = _h_.
 The graph opens up if a __ 0 and down if a __ 0.
Example 1
Graph a quadratic function in vertex form and find the maximum or minimum value.
Graph y =
1
(x + l)2  2.
2
GRAPH OF INTERCEPT FORM
y = a(x  p)(x  q):
Characteristics of the graph y = a(x  p)(x  q):
 The x-intercepts are _p_ and _q_.
 The axis of symmetry is halfway between ( _p , 0) and ( _q_ , 0). It has
pq
2
 The graph opens up if a _>_ 0 and opens down if a _<_ 0.
equation x =
Example 2
Graph a quadratic function in intercept form and find the maximum or minimum value.
Graph y = 2(x  1)(x  5).
Checkpoint Complete the following exercises.
1. Graph the function. Label the vertex and the axis of symmetry and find the maximum or minimum value.
y = (x  3)2 + 4
2. Graph the function. Label the vertex, axis of symmetry, and the x-intercepts and find the maximum or
minimum value.
y = (x  4)(x + 2)
FOIL METHOD
Words
To multiply two expressions that each contain two terms, add the products of the _First_ terms, the _Outer_
terms, the _Inner_ terms, and the _Last_ terms.
F
Example
O
I
L
(x + 4)(x + 7) = x2 + 7x + 4x + 28 = x2 + 11x + 28
Example 3
Change from intercept form to standard form
Write y = 3(x + 2)(x  5) in standard form.
Example 4
Change from vertex form to standard form
Write f(x) = 5(x + 2)2 + 8 in standard form.
Checkpoint Write the quadratic function in standard form.
3.
y = 4(x  3)2  10
4.
y = 3(x  7)(x + 6)
4.3 Solve x2  bx  c  0 by Factoring
Goal  Solve quadratic equations.
VOCABULARY
Monomial
An expression that is either a number, a variable, or the product of a number and one or more variables
Binomial
The sum of two monomials
Trinomial
The sum of three monomials
Quadratic equation
An equation in one variable that can be written in the form ax2 + bx + c = 0 where a  0
Root of an equation
A solution of a quadratic function
Zero of a function
The numbers p and q of a function in intercept form are also called the zeros of the function.
FACTOR x2  bx  c
STEP 1:
c
Example 1
STEP 2:
Find two numbers whose product is c
and whose sum is b.
b
c
r1
r2
b
Write in factored form.
( x  r1 )( x  r2 )
r1 r2  c
r1  r2  b
Factor trinomials of the form x2  bx  c
Factor the expression x2  7x  8.
SPECIAL FACTORING PATTERNS
Pattern Name
Difference of
Two Squares
a2  b2 = ( a + b )( a  b )
x2  4 = (x + 2)(x  2)
Perfect Square
Trinomial
a2 + 2ab + b2 = ( a + b )2
x2 + 6x + 9 = (x + 3)2
Perfect Square
Trinomial
a2  2ab + b2 = ( a  b )2
x2  4x + 4 = (x  2)2
Example 2
Factor with special patterns
Checkpoint Factor the expression. If it cannot be factored, say so.
1.
x2 + 7x + 12
2.
x2  81
ZERO PRODUCT PROPERTY
Words
If the _product_ of two expressions is zero, then _one_ or _both_ of the
expressions equals zero.
Algebra
If A and B are expressions and AB = _0_ , then A = _0_ or B = _0_ .
Example
If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = _5_ or
x = _2_ .
Example 3
Find the roots of an equation
Find the roots of the equation x2  2x  15 = 0.
Example 4
Find the zeros of a quadratic function
Find the zeros of the function y = x2  5x  6 by rewriting the function in intercept form.
Checkpoint Complete the following exercises.
3.
Find the roots of the equation x2  3x + 2 = 0.
4.
Find the zeros of the function y = x2 + 3x  40 by rewriting the function in intercept form.
4.4 Solve ax2 + bx + c = 0 by Factoring
Goal  Use factoring to solve equations of the form ax2 + bx + c = 0.
Step1:
Step 2:
Find two numbers whose product is c
ac
and whose sum is b.
ac
b
r1
r2
a
a
b
Write in factored form.
(ax  r1 )(ax  r2 )
Example 1
Factor ax2 + bx + c where c > 0
Factor 2x2 + 9x + 7.
Example 2
Factor ax2 + bx + c where c < 0
Factor 3x2  x  2.
r1 r2  c
r1  r2  b
Checkpoint Factor the expression. If it cannot be factored, say so.
1. 3x2 + 7x  20
2. 5x2  13x + 6
Example 3
Factor with special patterns
Factor the expression.
a. 16x2  36
b. 9y2 + 42y + 49
c. 25t2  110t + 121
Checkpoint Factor the expression.
3. 16y2  40y + 25
4. 4x2  81
Example 4
Factor out monomials first
Factor the expression.
a. 4x2  4
b. 3y2  18y
c. 4m2  10m + 24
d. 5z2  25z + 40
Checkpoint Factor the expression.
5. 36x2  16
6. 15p2 + 24p  63
Example 5
Solve quadratic equations
a.
2x2  x  21 = 0
b.
4r2  18r + 24 = 6r 12
Checkpoint Solve the equation.
7. 2x2 + 4x  30 = 0
8. z2 + 13z + 12 = 5z  4
4.5 Solve Quadratic Equations by Finding Square Roots
Goal  Solve quadratic equations by finding square roots.
VOCABULARY
Square root
A number r is a square root of a number s if r2 = s.
Radical
An expression of the form s where s is a number or expression
Radicand
The number s beneath the radical sign
Rationalizing the denominator
The process of eliminating a radical from the denominator of a fraction
Conjugates
The expressions a + b and a  b , used to rationalize the denominator, whose product is always a rational
number
PROPERTIES OF SQUARE ROOTS (a > 0, b> 0)
Example
Product Property
Quotient Property
a  ______
b
ab = _____
a =
b
Example 1
Use properties of square roots
a.
24
5
b. 9
64

18
a
b
18  9

2 3 2
2
2
2


25
25 5
Your Notes
Example 2
Rationalize denominators of fractions
4
.
Simplify (a) 7 and (b).
5 3
3
Checkpoint Simplify the expression.
1. 5
2.

10
9
11
Example 3
Solve a quadratic equation
Solve
1
(y  6)2 = 8.
4
Example 4
Model a dropped object with a quadratic function
Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take
for the water balloon to hit the sidewalk?
h = 16t2 + h0
Checkpoint Complete the following exercises.
3. Solve the equation 2x2  16 = 34.
4. In Example 4, suppose that the water balloon is dropped from a height of 27 feet. How long does it take for
the balloon to hit the sidewalk?
4.6 Perform Operations with Complex Numbers
Goal  Perform operations with complex numbers.
VOCABULARY
Imaginary unit i
The imaginary unit i is defined as i 
 1.
Complex number
A number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the
number bi is the imaginary part.
Imaginary number
A complex number a + bi where b  0
Complex conjugates
Two complex numbers of the form a + bi and a  bi
Complex plane
A coordinate plane where each point (a, b) represents a complex number a + bi. The horizontal axis is the real
axis and the vertical axis is the imaginary axis.
Absolute value of a complex number
The absolute value of a complex number z = a + bi, denoted z , is a nonnegative real number defined as
z   a 2  b 2 .
THE SQUARE ROOT OF A NEGATIVE NUMBER
Property
Example
3 
r  number,
i r.
If r is a positive real
1.
i
3
then
   r.
2. By Property (1), it follows that i r
2
i 3   i
2
2
3
Example 1
Solve a quadratic equation
2x2 + 15 = 35
Checkpoint Solve the equation.
1. 3x2 + 13 = 23
SUMS AND DIFFERENCES OF COMPLEX NUMBERS
To add (or subtract) two complex numbers, add (or subtract) their __real__ parts and their __imaginary__ parts
__separately__.
Sum of complex numbers:
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference of complex numbers:
(a + bi)  (c + di) = (a  c) + (b  d)i
Example 2
Add and subtract complex numbers
Write as a complex number in standard form.
a. (6 + 3i)  (4  /)
b. (2 + 5i) + (7  2i)
Example 3
Multiply complex numbers
Write the expression (2 + i)(5 + 2i) as a complex number in standard form.
.
Example 4
Divide complex numbers
Write the quotient
6  4i
2i
in standard form.
Checkpoint Write the expression as a complex number in standard form.
2. (12  2i) (16 + 3i)
3. 4i(9 + 5i)
4.
8  4i
3 i
5.
(4+4i) + ( 6+3i)
Example 5
Plot complex numbers
Plot the complex numbers in the same complex plane.
a. 4 + 3i
b. 5  4i
ABSOLUTE VALUE OF A COMPLEX NUMBER
The absolute value of a complex number z = a + bi, denoted | z |, is a _nonnegative_ real number defined as
| z | = a2  b2.
This is the distance of z from the _origin__ in the complex plane.
Example 6
Find absolute values of complex numbers
Find the absolute value of (a) 6  8i and (b) 6i.
Checkpoint Plot the complex numbers in the same complex plane. Then find the absolute value.
6. 4i
7.
8. 3+i
2+3i
4.7 Complete the Square
Goal  Solve quadratic equations by completing the square.
VOCABULARY
Completing the square
The process that allows you to write an expression of the form x2 + bx as the square of a binomial
COMPLETING THE SQUARE
2


Words To complete the square for the expression x + bx, add  b  .
 2 
2
b 
b
b 
Algebra x + bx +   =  x   x  
2 
2
____
 2 
2
2
b

=x 
2
_____
2
Example 1
Solving equations by square rooting.
x2  6 x  9  1
Checkpoint
Solve the equation by finding square roots.
1. x2  10 x  25  4
Example 2
Make a perfect square trinomial
Find the value of c that makes x2  12x  c a perfect square trinomial. Then write the expression as the
square of a binomial.
Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the
square of a binomial.
2. x2 24x + c
3. x2 + 10x +c
Example 3
Solve ax2  bx  c = 0 when a = 1
Solve x2  10x  13  0 by completing the square.
Checkpoint Solve the equations by completing the square.
4. x2  8x + 7 = 0
Example 4
Write a quadratic function in vertex form
Write y = x2  14x  44 in vertex form. Then identify the vertex.
Checkpoint Write the quadratic function in vertex form. Then identify the vertex.
5. y = x2  12x + 22
6. y = x2 + 16x +53
Example 5 (Day 2)
Solve ax2  bx  c  0 when a  1
Solve 3x2  12x  27 = 0 by completing the square.
Checkpoint Solve the equations by completing the square.
5. 2x2  20x + 24 = 0
4.8 Use the Quadratic Formula and the Discriminant
Goal  Solve quadratic equations using the quadratic formula.
VOCABULARY
Quadratic formula
The formula that gives the solutions to any quadratic equation
Discriminant
The expression b2  4ac under the radical sign of the quadratic formula
THE QUADRATIC FORMULA
Let a, b, and c be real numbers such that a  0. The solutions of the quadratic equation ax2 + bx + c are:
b 
x=
b 2  4 ac
2 a
Example 1
Solve an equation with two real solutions
Solve x2  7x  6.
Example 2
Solve an equation with one real solution
Solve 2x2  8x  8 = 0.
Checkpoint Use the quadratic formula to solve the equation.
1.
2x2 + 12x = 16
2.
4x2  13x = 7x  25
3.
3x2  6x + 6 = 0
4.
x2  3x + 3 = 0
USING THE DISCRIMINANT OF ax2 + bx + c = 0
When b2  4ac > 0, the equation has _two real solutions_. The graph has _two_x-intercepts.
When b2  4ac = 0, the equation has _one real solution_. The graph has _one_x-intercept.
When b2  4ac < 0, the equation has _two imaginary solutions_. The graph has _no_x-intercepts.
Example 4
Use the discriminant
Find the discriminant of the quadratic equation and give the number and type of solutions of the
equation.
a. x2 + 6x + 5 = 0
b. x2 + 6x + 9 = 0
c. x2 + 6x + 13 = 0
Checkpoint Find the discriminant of the quadratic equation and give the number and type of solutions
of the equation.
5. x2  8x + 17 = 0
6.
x2 + 4x + 3 = 0
7.
x2 + 2x  1 = 0
8.
x2 + 6x + 4 = 0