Download Day 57 - 61 EOC Quadratics Reivew

Name:_______________________________
Period:____
Chapter 4 Notes Packet on Quadratic Functions and Factoring
Notes #15: Graphing quadratic equations in standard form, vertex form, and intercept form.
A. Intro to Graphs of Quadratic Equations: y  ax 2  bx  c

A ____________________ is a function that can be written in the form y  ax 2  bx  c where
a, b, and c are real numbers and a  0. Ex: y  5 x 2 y  2 x 2  7 y  x 2  x  3

The graph of a quadratic function is a U-shaped curve called a ________________. The
maximum or minimum point is called the _____________
Identify the vertex of each graph; identify whether it is a minimum or a maximum.
1.)
2.)
Vertex: (
,
) _________
Vertex: (
3.)
) _________
,
) _________
4.)
Vertex: (
,
) _________
Vertex: (
B. Key Features of a Parabola:

,
y  ax 2  bx  c
Direction of Opening: When a  0 , the parabola opens ________:
When a  0 , the parabola opens ________:

Width: When a  1 , the parabola is _______________ than y  x 2
When a  1 , the parabola is the ________ width as y  x 2
When a  1 , the parabola is ________ than y  x 2

Vertex: The highest or lowest point of the parabola is called the vertex, which is on the axis of
symmetry. To find the vertex, plug in x 

b
and solve for y. This yields a point (____, ____)
2a
Axis of symmetry: This is a vertical line passing through the vertex. Its equation is: x 
b
2a

x-intercepts: are the 0, 1, or 2 points where the parabola crosses the x-axis. Plug in y = 0 and
solve for x.

y-intercept: is the point where the parabola crosses the y-axis. Plug in x = 0 and solve for y.
1
Without graphing the quadratic functions, complete the requested information:
5
5.) f ( x)  3x 2  7 x  1
6.) g ( x)   x 2  x  3
4
What
is
the
direction
of opening? _______
What is the direction of opening? _______
Is the vertex a max or min? _______
Is the vertex a max or min? _______
2
Wider or narrower than y = x2 ? ___________
Wider or narrower than y = x ? __________
7.)
y
8.) y  0.6 x 2  4.3x  9.1
2 2
x  11
3
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? __________
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
The parabola y = x2 is graphed to the right.
Note its vertex (___, ___) and its width.
You will be asked to compare other parabolas to
this graph.
C. Graphing in STANDARD FORM ( y  ax 2  bx  c ): we need to find the vertex first.
Vertex
Graphing
- list a = ____, b = ____, c = ____
- put the vertex you found in the center of
your
x-y chart.
b
- find x =
- choose 2 x-values less than and 2 x-values more
2a
than
your vertex.
- plug this x-value into the function (table)
- this point (___, ___) is the vertex of the parabola - plug in these x values to get 4 more points.
- graph all 5 points
Find the vertex of each parabola. Graph the function and find the requested information
9.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____
y
10
9
8
7
6
5
4
3
Vertex: _______
Max or min? _______
Direction of opening? _______
Axis of symmetry: ________
Compare to the graph of y = x2
_________________________
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
2
10.) h(x) = 2x2 + 4x + 1
y
10
9
8
7
Vertex: _______
Max or min? _______
Direction of opening? _______
Axis of symmetry: ________
Compare to the graph of y = x2
_________________________
6
5
4
3
2
1
x
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
11.) k(x) = 2 – x –
1 2
x
2
y
10
9
8
7
6
Vertex: _______
Max or min? _______
Direction of opening? _______
Axis of symmetry: ________
Compare to the graph of y = x2
_________________________
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
12.) State whether the function y = 3x2 + 12x  6 has a minimum value or a maximum
value. Then find the minimum or maximum value.
1 2
x  5 x  7 . State whether it is a minimum or maximum. Find that
2
minimum or maximum value.
13.) Find the vertex of y 
3
Another useful form of the quadratic function is the vertex form: ________________________________.
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 ___________ h units and ___________ k
units.
 The vertex is (___, ___).
 The axis of symmetry is x = ___.
 The graph opens up if a ___ 0 and down if a ___ 0.
Find the vertex of each parabola and graph.
2
13.) y   x  1  2
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
Vertex: _______
-7
-8
-9
-10
1
2
14.) y    x  1  5
3
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
Vertex: _______
-7
-8
-9
-10
15.) Write a quadratic function in vertex form for the function whose graph has its vertex
at (-5, 4) and passes through the point (7, 1).
4
GRAPH OF INTERCEPT FORM y = a(x  p)(x  q):
Characteristics of the graph y = a(x  p)(x  q):

The x-intercepts are ___ and ___.

The axis of symmetry is halfway between (___, 0) and ( ___ , 0)
and it has equation x =

2
The graph opens up if a ___ 0 and opens down if a ___ 0.
16.) Graph y = 2(x  1)(x  5)
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
x-intercepts: _______, _______
-6
-7
-8
-9
Vertex: _______
-10
Converting between forms:
From intercept form to standard form

Use FOIL to multiply the binomials
together

Distribute the coefficient to all 3 terms
From vertex form to standard form

Re-write the squared term as the
product of two binomials

Use FOIL to multiply the binomials
together

Distribute the coefficient to all 3 terms

Add constant at the end
Ex:
y  2  x  5 x  8
Ex: f  x   4  x  1  9
2
5
Notes 16: Sections 4-3 and 4-4: Solving quadratics by Factoring
A. Factoring Quadratics
Examples of monomials:_______________________________
Examples of binomials:________________________________
Examples of trinomials:________________________________
Strategies to use: (1) Look for a GCF to factor out of all terms
(2) Look for special factoring patterns as listed below
(3) Use factor by grouping
(4) Check your factoring by using multiplication/FOIL
Factor each expression completely. Check using multiplication.
1.) 3 x 2  15 x
2.) 6 x 2  24
3.) x 2  5 x  24
4.) 25 x 2  81
6.) 4 x 2  12 x  9
5.) m2  22m  121
6
7.) 5 x 2  17 x  6
8.) 3x 2  5 x  12
9.) 25t2  110t + 121
10.) 16 x 2  36
11.) 9a 2  42a  49
12.) 6 x 2  33x  36
B. Solving quadratics using factoring
To solve a quadratic equation is to find the x values for which the function is equal to _____. The
solutions are called the _____ or _______of the equation. To do this, we use the Zero Product
Property:
Zero Product Property
List some pairs of numbers that multiply to zero:
(___)(___) = 0
(___)(___) = 0
(___)(___) = 0
(___)(___) = 0
What did you notice? _______________________________________________
ZERO PRODUCT PROPERTY
If the _________ of two expressions is zero, then _______ or _______ of
the expressions equals zero.
Algebra
If A and B are expressions and AB = ____ , then A = _____ or B = __.
Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is,
x = __________ or x = _________.
Use this pattern to solve for the variable:
1.
get the quadratic = 0 and factor completely
2.
set each ( ) = 0 (this means to write two new equations)
3.
solve for the variable (you sometimes get more than 1 solution)
7
Find the roots of each equation:
13.) x 2  7 x  30  0
15.) 3x 2  2 x  21
6  4
8
2
14.)  x   x    0
7  5
9
3
Find the zeros of each equation:
17.) v(v + 3) = 10
16.) 2 x 2  8 x  30  x  34
18.) 2 x 2  x  15
Find the zeros of the function by rewriting the function in intercept form:
19.) y  6 x 2  3x  63
20.) f  x   12x2  6x  6
21.) g  x   49x2 16
Graph the function. Label the vertex and axis of symmetry:
22.) y  2  x  1 ( x  3)
y
10
9
8
7
6
5
4
Vertex: _______
Maximum or minimum value: _______
x-intercepts: _______
Axis of symmetry: ________
Compare width to the graph of y = x2
_________________________
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
8
D. Solving Quadratic Equations Using Square Roots




Isolate the variable or expression being squared (get it ______________)
Square root both sides of the equation (include + and – on the right side!)
This means you have _____________ equations to solve!!
Solve for the variable (make sure there are no roots in the denominator)
8.) x2 = 25
10.)
4x2 – 1 = 0
12.) (2y + 3)2 = 49
9.) 3x2 = 81
11.)
m2
3  2
15
13.) 3  x  2   6  34
2
9
Section 4.7: Completing the Square
B. Review: Solving Using Square Roots
 Factor and write one side of the equation as the square of a binomial
 Square root both sides of the equation (include + and – on the right side; 2 equations!
 Solve for the variable (make sure there are no roots in the denominator)
1) (k + 2)2 = 12
2.) x2 + 2x + 1 = 8
3.) n2 – 14n + 49 = 3
C. Completing the Square ax 2  bx  c  0

Take half the b (the x coefficient)

Square this number (no decimals – leave as a fraction!)

Add this number to the expression

Factor – it should be a binomial, squared (
)2
4.) x2 + 6x + _____
(
)(
(
5.) m2 – 14m + _______
)
)2
Find the value of c such that each expression is a perfect square trinomial. Then write the expression as
the square of a binomial.
6.) w2 + 7w + c
7.) k2 – 5k + c
Solving by Completing the Square:

Collect variables on the left, numbers on the right

Divide ALL terms by a; leave as fractions (no decimals!)

Complete the square on the left – add this number to BOTH sides

Square root both sides (include a ______ and _______ equation!)

Solve for the variable
8.) x2 + 4x – 5 = 0
9.) m2 – 5m + 11 = 10
10
10.) 2k 2  9k  3  k  17
11.) 3w2  4w  11  2  2 w
12.) 2x2 – 3x – 1 = 0
13.) 3x 2  x  2 x  6
Cumulative Review: Solving Quadratics
Solve by factoring:
14.) 12k2 – 5k = 2
15.) 49m2 – 16 = 0
Solve by using square roots:
16.) 4w2 = 18
17.) 3y2 – 8 = 0
11
Notes #19: Sec. 4-8 Use the Quadratic Formula and the Discriminant
A. Review of Simplifying Radicals and Fractions

Simplify expression under the radical sign; reduce

Reduce only from ALL terms of the fraction.
(You can’t reduce a number outside of a radical with a number inside of a radical)

Make sure that you have TWO answers
Simplify:
6  18
1.)
2
2.)
5  20
2
3.)
4  20
4
4.)
8  27
2
5.)
9  (5)2  (5)(2)(3)
4
6.)
9  (6)2  4(3)(3)
4
B. Solving Quadratics using the Quadratic Formula
So far, we have solved quadratics by: (1) _______________, (2) ______________, and
(3) ___________________.
The final method for solving quadratics is to use the quadratic formula.
Solving using the quadratic formula:

Put into standard form (ax2 + bx + c = 0)

List a = , b = , c =
b  b2  4ac
x
2a

Plug a, b, and c into

Simplify all roots (look for
12
1  i ); reduce
Solve by using the quadratic formula:
2
1.)
x + x = 12
x
b  b2  4ac
2a
(std. form):
a = _____
b = _____
c = _____
2.) 5x – 8x = -3
2
b  b2  4ac
x
2a
(std. form):
a = _____
b = _____
c = _____
3.) -x2 + x = -1
4.) 3x2 = 7 – 2x
5.) -x2 + 4x = 5
6.) 4( x  1)2  6 x  2
13
C. Using the Discriminant
 Quadratic equations can have two, one, or no solutions (x-intercepts). You can determine how
many solutions a quadratic equation has before you solve it by using the ________________.
b  b2  4ac
 The discriminant is the expression under the radical in the quadratic formula: x 
2a
Discriminant = b2 – 4ac
If b2 – 4ac < 0, then the equation has 2 imaginary solutions
If b2 – 4ac = 0, then the equation has 1 real solution
If b2 – 4ac > 0, then the equation has 2 real solutions
A. Finding the number of x-intercepts
Determine whether the graphs intersect the x-axis in zero, one, or two points.
1.) y  4 x 2  12 x  9
2.) y  3x 2  13x  10
B. Finding the number and type of solutions
Find the discriminant of the quadratic equation and give the number and type of solutions of the
equation.
3.) 3 x 2  5 x  1
4.) x 2  3 x  7
5.) 9x2 – 6x = 1
6.) 4x2 = 5x + 3
14
Cumulative Review Problems:
Solve by factoring:
7.) 4m2 +5m – 6 = 0
8.) 3x3 – 27x = 0
Solve by using square roots:
9.) 4b2 + 1 = 0
10.) (3x + 1)2 = 18
Solve by completing the square:
11.) 4m2 + 12m +5 = 0
12.) x2 – 7x – 18 = 0
For #13, find the vertex of the parabola. Graph the function and find the requested information
13.) g(x) = -2x2 + 8x – 5
y
10
9
8
7
6
5
Vertex: _______
4
3
2
Max or min value: _______
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Direction of opening? _______
-2
-3
Compare width to y = x2 :
-4
-5
-6
_______________
-7
-8
-9
Axis of symmetry: ________
-10
15
Notes #20: Section 4-9 Graph and Solve Quadratic Inequalities
GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES
To graph a quadratic inequality, follow these steps:
Step 1 Graph the parabola with equation y = ax2+ bx + c.
Make the parabola _______________ for inequalities with < or > and ______________ for inequalities
with  or .
Step 2 Test a point (x, y) ______________ the parabola to determine whether the point is a solution of the
inequality.
Step 3 Shade the region ______________ the parabola if the point from Step 2 is a solution. Shade the
region ________________ the parabola if it is not a solution.
1.) y  x2  2x + 3
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
2.) y ≥ x2  3x – 4
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
16
1
2
3
4
5
6
7
8
9 10
3.) y > 3x2 +3x 5
y  x2 +5x + 10
4.) y  -x2 +4
y  x2 − 2x− 3
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
17
1
2
3
4
5
6
7
8
9 10
Notes# 21: Sec. 4-10 Write Quadratic Functions and Models
A. When given the vertex and a point
 Plug the vertex in for (h, k) in y  a( x  h)2  k
 Plug in the given point for (x, y)
 Solve for a. Plug in a, h, k into y  a( x  h)2  k
1.) Write a quadratic equation in vertex form
for the parabola shown.
2.) Write a quadratic function in vertex form for the function whose graph has its vertex at (2, 1) and
passes through the point (4, 5).
B. When given the x-intercepts and a third point
 Plug in the x-intercepts as p and q into y = a(x  p)(x  q)
 Plug in the given point for (x, y)
 Solve for a. Plug in a, h, k into y = a(x  p)(x  q)
3.) Write a quadratic function in intercept form
for the parabola shown.
18
Notes #22: Chapter 4 Review
To graph a quadratic function, you must FIRST find the vertex (h, k)!!
(A) If the function starts in standard form y  ax 2  bx  c :
b
1st: The x-coordinate of the vertex, h, =
2a
2nd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function &
solving for y.
(B) If the function starts in intercept form y  a( x  p)( x  q ) :
1st: Find the x-intercepts by setting the factors with x equal to 0 & solving for x.
2nd: The x-coordinate of the vertex is half way between the x-intercepts.
3rd: Find the y-coordinate of the vertex, k, by plugging the x-coordinate into the function &
solving for y.
(C) If the function starts in vertex form y  a( x  h)2  k :
1st: pick out the x-coordinate of the vertex, h. REMEMBER: h will have the OPPOSITE sign
as what is in the parenthesis!!
nd
2 : Pick out the y-coordinate of the vertex, k. It will have the SAME sign as the what is in the
equation!
Direction of Opening:
If a is positive, the graph opens up
If a is negative, the graph opens down.
Width of the function:
If a  1 , the graph is narrower than y  x 2
If a  1 , the graph is wider than y  x 2
19
Graph each function. (A) State the vertex. (B) State the direction of opening (up/down). (C) State
whether the graph is wider, narrower, or the same width as y  x 2 .
1
1.) f ( x)  ( x  6) 2  5
2
y
10
9
8
7
6
Vertex: _______
5
4
3
Direction of opening? _______
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Compare width to the graph of y = x2
_________________________
-2
-3
-4
-5
-6
-7
-8
-9
-10
2.) k(x) = x2 + 2x + 1
y
10
9
8
7
6
5
4
Vertex: _______
3
2
1
Direction of opening? _______
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
Compare width to the graph of y = x2
_________________________
-4
-5
-6
-7
-8
-9
-10
3.) f(x) = x2 – x – 6
y
10
9
8
7
6
Vertex: _______
5
4
3
Direction of opening? _______
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Compare width to the graph of y = x2
_________________________
-2
-3
-4
-5
-6
-7
-8
-9
-10
20
4.) f ( x)   x2  2 x  3
y
10
9
8
7
Vertex: _______
6
5
4
3
Direction of opening? _______
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
Compare width to the graph of y = x2
_________________________
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
5.) h( x)  2 x 2  4 x  1
y
10
9
8
7
Vertex: _______
6
5
4
3
Direction of opening? _______
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Compare width to the graph of y = x2
_________________________
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
6.) g ( x)   ( x  4)( x  6)
3
y
10
9
8
Vertex: _______
7
6
5
4
Direction of opening? _______
3
2
1
Compare width to the graph of y = x2
_________________________
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
21
Methods for Solving Quadratic Equations:
A.) Factoring
1st: Set equal to 0
2nd: Factor out the GCF
3rd: Complete the X & box method to find the factors
4th: Set every factor that contains an x in it, equal to 0 & solve for x.
B.) Completing the Square
1st: Move the constant (number with no variable) to the right so that all variables are on the left
& all constants are on the right.
nd
2 : Divide every term in the equation by the value of a, if it is not already 1.
2
b
3rd: Create a perfect square trinomial on the left side by adding   to both sides.
2
2
b

4 : Factor the left side into a  x   and simplify the value on the right side.
2

th
5 : Take the square root of both sides of the equation. REMINDER: Don’t forget the 
6th: Solve for x
th
C.) Finding Square Roots
1st: Isolate the term with the square.
2nd: Take the square root of both sides of the equation. REMINDER: Don’t forget the 
3rd: Solve for x.
D.) Quadratic Formula
1st: Set the equation equal to 0.
2nd: Find the values of a, b, and c & plug them into the Quadratic Formula:
b  b2  4ac
2a
rd
3 : Simplify the radical as much as possible.
4th: If possible, simplify the numerator into integers.
x
46 3
), divide BOTH
2
terms by the number in the denominator (the example would result in 2  3 3 )
5th: Divide. REMINDER: If you have 2 terms in the numerator (ex:
Examples: Solve each equation by the method stated.
By Square Roots:
1.) (2y + 3) 2 = 49
2.) (3m – 1) 2 = 20
22
3.) (3r  5)2  1  11
By Factoring:
4.) x2 – 4x – 5 = 0
5.) 3 x 2  2 x  21
6.) 2 x 2  x  15
By Completing the Square:
7.) x2 – 6x – 11 = 0
8.) 2y2 + 6y – 18 = 0
9.) 2x2 – 3x – 1 = 0
By Quadratic Formula:
10.) x2 + x = 12
11.) 5x2 – 8x = -3
12.) 2x2 = 4 – 7x
23
Chapter 4 Review Sheet
Please complete each problem on a separate sheet of paper. Show all of your work and please use
graph paper for all graphs.
For questions 1-3, solve by factoring.
1. x2 12 x  32  0
2. 3x2  8x  5  0
3. 2x2  4x  30  0
For questions 4-5, solve by finding square roots.
1
2
4. ( x  1) 2  16  0
5. 2  x  1  3  6
4
For questions 6-8, solve by completing the square.
6. x2 16x  15  0
1
8.  x 2  4 x  6  0
2
7. 5x2  10 x  20  0
For questions 9-10, solve by using the quadratic formula.
9.  x2  3x  5  0
10. 3x2  2 x  x2  5x 1
For questions 11-13, simplify each expression.
48  32
5
11.
12.
4
48
13.
252  3
For questions 14-16, write each function in vertex form, graph the function, and label the vertex
and axis of symmetry.
1
14. y  x 2  16 x  2
15. y  2 x 2  4 x  7
16. y  ( x  4) 2  8
3
For question 17-19, graph each inequality.
17. y  2( x  1)2  5
18. y   x 2  8 x  2
19. y  2 x 2  12 x  16
For questions 20-21, write a quadratic function in vertex form whose graph has the given vertex
and passes through the given point.
20. Vertex (2, 3) and passes through (-3, 7)
21. Vertex (-1, -5) and passes through (2, -1)
For questions 22-23, write a quadratic function in standard form whose graph passes through
the given points.
22. (1, 1), (0, -2), (2, 8)
23. (-1, -7), (1, -5), (2, -1)
For questions 24-28, write the expression as a complex number in standard form.
24. (7  3i)  (9  4i)
25. (6  2i)(8  3i)
26. (3  7i )  (9  2i)
27.
2  5i
7i
28.
4  7i
3i
Chapter 4 Test and Notes Check Tomorrow!
24