Download CHAPTER 8: POLYNOMIALS AND FACTORING

Name:
Date:__________
Period: __________
CHAPTER 9: Quadratic Equations and Functions
Notes #23
9-1: Exploring Quadratic Graphs
A. Graphing y  ax 2

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.
Examples: y  5 x 2


y  x2  x  3
The graph of a quadratic function is a U-shaped curve called a ________________. When
graphed it will look like:

y  x2  7
OR
You can fold a parabola so that the two sides match exactly. This property is called:
_____________.
The highest or lowest point of the parabola is called the ________________, which is on the
axis of symmetry.
B. Identifying a Vertex
Identify the vertex of each graph. Tell whether it is a minimum or a maximum.
1.)
Vertex: (
2.)
,
) _________
3.)
Vertex: (
Vertex: (
,
) _________
,
) _________
4.)
,
) _________
Vertex: (
Algebra 1 Notes: Chapter 9
-2Graph each function. State the domain, the vertex (min/max point), the range, the x-intercepts,
and the axis of symmetry.
5.) f(x)= x2 – 4
y
10
9
8
7
6
5
4
3
2
1
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
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
6.) h(x) = -2x2
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
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
-2
-3
-4
-5
-6
-7
-8
-9
-10
7.) f(x) =
1 2
x
2
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
1
2
3
4
5
6
7
8
9 10
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
Algebra 1 Notes: Chapter 9
-38.) k(x) = x2 + 2x + 1
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
3
4
5
6
7
8
9 10
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
-2
-3
-4
-5
-6
-7
-8
-9
-10
9.) f(x) = x2 – x - 6
y
10
9
8
7
6
5
4
3
2
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
-2
-3
-4
-5
-6
-7
-8
-9
-10
C. Comparing Widths of parabolas
The value of a, the coefficient of the x 2 term in a quadratic function, affects the width of the parabola
as well as the direction in which it opens.

When a  1, then the parabola is steeper, (or _________) than y = x2

When a  1, then the parabola is not as steep, (or _________) than y = x2
Order each group of quadratic functions from widest to narrowest graph:
10.) f ( x)  3x 2 , f ( x)  4 x 2 , f ( x) 
1 2
x
2
5
11.) y  2 x 2 , y   x 2 , y   x 2
4
Algebra 1 Notes: Chapter 9
-4D. Applications
12.) A monkey drops an banana from a branch 64 feet above the ground. Gravity causes the banana to
fall. The function h  16t 2  64 gives the height of the banana, h, in feet, after t seconds.
a) Graph this quadratic function
b) When does the banana hit the ground?
t
h(t )  16t 2  64
(t, h(t))
70
60
50
feet
40
30
20
10
1
2
3
time (sec)
13) A bungee jumper dives from a platform. The function h = -16t2 + 160 describes her height, h,
after t seconds in the air.
a) What will her height be after 1 second?
b) what will her height be after 2 seconds?
c) How far did she fall between 1 and 2 seconds in the air?
Algebra 1 Notes: Chapter 9
-5Notes #24
9-2: Quadratic Functions
y = ax2 + bx + c
One key characteristic of a parabola is its vertex (min/max point). Yesterday we found the vertex after
we graphed the function. It would help to find the vertex first.
Vertex
- find x =
b
2a
- plug this x-value into the function (table)
- this point (___, ___) is the vertex of the parabola
Graphing
- put the vertex you found in the center of
your x-y chart.
- choose 2 x-values less than and 2 x-values more
than your vertex.
- 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
1.) f(x)= -x2 + 2x + 3
a = ____, b = ____, c = ____
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
Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
2.) h(x) = 2x2 + 4x + 1
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
1
2
3
4
5
6
7
8
9 10
Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
Algebra 1 Notes: Chapter 9
-63.) k(x) = 2 – x –
1 2
x
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
-7
-8
-9
-10
Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
Without graphing the quadratic functions, complete the requested information:
4.) f ( x)  3x 2  7 x  1
5
5.) g ( x)   x 2  x  3
4
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 ? ___________
6.)
y
2 2
x  11
3
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
7.) y  0.6 x 2  4.3x  9.1
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
B. Application
8.) Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an
initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How
far above the ground will it be? The equation h  16t 2  88t  520 gives the star’s height h in feet at
time t in seconds.
Algebra 1 Notes: Chapter 9
-7Notes #25
9-3: Finding and Estimating Square Roots
A. Finding Square Roots

The expression __________ means the positive, or __________ square root.

The expression __________ means the negative square root.

The expression ___________ means both the ____________ and _____________ square root
Simplify each expression.
1.)
4.)  0
5.)
7)
8.)  72
27
10.) 
3.) 
2.)  100
64
4
5
6.)  0.09
25
11.) 
9
16
9.) 108
45
4
12.)
2.25
B. Rational and Irrational Square Roots
Tell whether each expression is rational or irrational.
13.)  144
14.) 
1
5
17.) Between what two consecutive integers is
15.)
28.34 ?
1
9
16.)
7
Algebra 1 Notes: Chapter 9
-818.) Between what two consecutive integers is
68.7 ?
19.) Between what two consecutive integers is  14.3 ?
C. Application: Pythagorean Theorem (Review)
Use the Pythagorean theorem (_________________) to solve for the missing side of the right
triangle.
20.)
21.)
6
4
8
x
y
4
9-4: Solving Quadratic Equations
A. Solving Quadratic Equations by Graphing

The solutions of a quadratic equation and the related x-intercepts are often called _______ of
the equation or _______ of the function.
1.) The function f(x) = x2 + x – 6 is graphed to the left.
a) Circle and name the zeros of the function graphed here.
(
,
) and (
,
)
b) Use this graph to solve the equation: x2 + x – 6 = 0
(This is asking: “At what x-values does y = 0?”)
Algebra 1 Notes: Chapter 9
-9B. Solving by Graphing
Solve each equation by graphing the related function:

Find the vertex and 4 other points on the parabola; graph.

Find the x-intercepts from the graph. These are the _______ or _______.
2.) x2 – 4 = 0
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
1
2
3
4
5
6
7
8
9 10
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
3.) 2x2 – 2 = 0
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
4.) x2 + 6 = 0
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
1
2
3
4
5
Algebra 1 Notes: Chapter 9
- 10 C. 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)
5.) x2 = 25
6.) 3x2 = 48
7.)
4x2 – 1 = 0
8.) 3m2 – 5 = 0
9.)
2y2 – 81 = 0
10.) 36b2 – 7 = 0
11.)
(x – 1)2 = 4
12.) (2y + 3)2 = 49
13.) (r + 5)2 = 12
14.) (3m – 1)2 = 20

Algebra 1 Notes: Chapter 9
- 11 If the left side is not already factored or squared, _______________ it!
15.) x2 + 2x + 1 = 8
16.) n2 – 14n + 49 = 3
17.) w2 + 22w + 121 = 169
18.) g2 + 10g + 25 = 18
D. Application
19.) A museum is planning an exhibit that will contain a large globe. The surface area of the globe will
be 100 ft2. Find the radius of the sphere producing this surface area. Use the equation S  4 r 2 ,
where S is the surface area and r is the radius.
Algebra 1 Notes: Chapter 9
- 12 Notes #26
9-5: Solving Quadratic Equations by Factoring
A. Solving Quadratic Equations
Zero Product Property

List some pairs of numbers that multiply to zero:
(___)(___) = 0

(___)(___) = 0
(___)(___) = 0
(___)(___) = 0
What did you notice? _______________________________________________
Use this pattern to solve for the variable:
1. get = 0 and factor (sometimes this is done for you)
2. set each ( ) = 0 (this means to write two new equations)
3. solve for the variable (you sometimes get more than 1 solution)
1.) (3)(x) = 0
2.) (2)(x + 1) = 0
4.) (m + 1)(5m – 3) = 0
5.)
7.) x2 – 4x – 5 = 0
8.) y2 + 6 + 5y = 0
3
w  2w  9   0
5
3.) -2y(y – 7) = 0
6  4
8
2
6.)  x   x    0
7  5
9
3
9.) 4v2 – 9 = 0
10.) x 2  x  42  0
13.) v(v + 3) = 10
11.) 3x 2  2 x  21
Algebra 1 Notes: Chapter 9
- 13 2
12.) 2 x  x  15
14.) b(b – 2) = 3(b + 2)
B. Solving Word Problems with Quadratics
Steps:
1. Draw a picture and define your variable (let statement)
2. Write an equation
3. Get = 0 (bring all variables and numbers to one side)
4. Factor completely and solve
5. Do all the answers make sense?
6. Write your answer in a complete sentence
Translate and solve:
15.) The square of a positive number minus twice the number is 48. Find the number.
Let n = _____________
_________ - _________ = ______
16.) One more than a negative number times one less than that number is 8. Find the number.
Let n = ______________
(_________)(_________) = _____
Algebra 1 Notes: Chapter 9
- 14 17.) The product of two consecutive integers is 12. Find the integers.
Let x = 1st integer
______ = 2nd integer
18.) The product of two consecutive odd integers is 35. Find the integers.
Let x = 1st odd integer
______ = 2nd odd integer
19.) The length of a rectangle is 3ft greater than its width. The area of the rectangle is 54ft2. Find the
length and the width of the rectangle.
20.) The area of a square is 5 more square inches than there are inches in the square’s perimeter. Find
the length of a side of the square.
Algebra 1 Notes: Chapter 9
- 15 21.) Two less than the square of a number is equal to the number. Find the number.
___________ - _________ = _______
22.) The sum of the square of a number and three times the number is the same as one less than the
number. Find the number.
__________ + _________ = _________ - _________
Notes #27
9-6: Completing the Square
So far in this course, we have solved quadratics by _______________, __________________ and
___________________. We will eventually learn two more ways to solve quadratics.
Solve these equations. What makes these quadratics “easy” to solve?
a) (x – 1)2 = 9
b) (k + 2)2 = 12
Solving quadratics by _________________ ______ _______________ helps us turn all quadratics
into this form.
Complete the square:
 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
1.) x2 + 6x + _____
(
)(
(
)2
2.) m2 – 14m + _______
)
Algebra 1 Notes: Chapter 9
- 16 Find the value of n such that each expression is a perfect square trinomial:
3.) w2 + 7w + n
4.) k2 – 5k + n
5.) j2 – j + n
6.) y2 + 18y + n
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 (simplify all roots)
7.) x2 + 4x – 5 = 0
8.) x2 – 6x – 11 = 0
9.) k2 – 4k – 7 = 0
10.) m2 – 5m + 1 = 0
11.) 2y2 + 6y – 18 = 0
12.) 2x2 – 3x – 1 = 0
Worksheet DA #29: Cumulative Review: Solving Quadratics
Algebra 1 Notes: Chapter 9
- 17 Name__________________
Solve by factoring:
1.) 12k2 – 5k = 2
2.) 49m2 – 16 = 0
Solve by using square roots:
3.) 4w2 = 18
4.) 3y2 – 8 = 0
5.) 5m2 – 16 = 0
6.) (2x – 1)2 = 20
Solve by completing the square:
7.) x2 – 10x + 7 = 0
8.) 4m2 + 12m – 7 = 0
9.) 3y2 – 2y – 1 = 0
10.) 2x2 – 20x = -50
11.)
5x2 + 13x + 7 = 0
12.) ax2 + bx + c = 0
Algebra 1 Notes: Chapter 9
- 18 -
Notes #28
9-7: Using the Quadratic Formula
A. Review of Simplifying Radicals and Fractions
 Simplify expression under the radical sign, reduce
 Reduce only from ALL terms of the fraction
1.)
6  18
2
2.)
5  20
2
3.)
4  20
4
4.)
8  27
2
9  (5)2  (4)(2)(3)
5.)
4
9  (6)2  4(3)(3)
6.)
4
Algebra 1 Notes: Chapter 9
- 19 B. Solving Quadratics using the Quadratic Formula
So far, we have solved quadratics by: (1) _______________, (2) ______________,
(3) ___________________, and (4) _________________
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, reduce
Solve by using the quadratic formula:
1.)
x2 + x = 12
b  b2  4ac
x
2a
(std. form):
a = _____
b = _____
c = _____
2.) 5x2 – 8x = -3
(std. form):
a = _____
b = _____
c = _____
b  b2  4ac
x
2a
Algebra 1 Notes: Chapter 9
- 20 3.) 2x2 = 4 – 7x
4.) 3x2 – 8 = 10x
5.) -x2 + x = -1
6.) 3x2 = 7 – 2x
Review of Solving Quadratics:
Solve by factoring:
7.) 4m2 +5m – 6 = 0
8.) 3x3 – 27x = 0
Solve by using square roots:
9.) 4b2 – 1 = 0
10.) 3y2 – 36 = 0
11.) (3x + 1)2 = 18
Algebra 1 Notes: Chapter 9
- 21 Solve by completing the square:
12.) x2 – 10x + 9 = 0
12.) x2 – 7x – 18 = 0
13.) 4m2 + 12m + 5 = 0
14.) 3y2 + 2y – 1 = 0
Solve by using the quadratic formula:
15.) x2 – 20 = 0
16.)
x2 – 6x + 9 = 0
Algebra 1 Notes: Chapter 9
- 22 Notes #29
9-8: Using the Discriminant

Quadratic equations can have two, one, or no solutions. You can determine how many solutions
a quadratic equation has before you solve it by using the ________________.

The discriminant is the expression under the radical in the quadratic formula:
x
b  b2  4ac
2a
Discriminant = b2 – 4ac
If b2 – 4ac = 0, then the equation has 1 solution
If b2 – 4ac < 0, then the equation has 0 real solutions
If b2 – 4ac > 0, then the equation has 2 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 of solutions
Find the number of solutions for the following:
3.) 3 x 2  5 x  1
4.) x 2  3 x  7
5.) 9x2 – 6x = 1
6.) 4x2 = 5x + 3
Algebra 1 Notes: Chapter 9
- 23 -
C. Review of Solving Quadratics
Solve by factoring:
7.) 2x2 + 12x = -10
8.) 16(x – 1) = x(x + 8)
Solve by using square roots:
9.) 3b2 – 1 = 7
10.) (3x + 1)2 = 18
Solve by completing the square:
11.) x2 – 10x – 11 = 0
12.) x2 – 3x – 6 = 0
Algebra 1 Notes: Chapter 9
- 24 Solve using the quadratic formula:
13.)
14.) x2 = 8 – 6x
6x2 + 7x = 5
Graph the quadratic. Name the vertex, axis of symmetry, x-intercepts, domain, and range.
15.) f(x)= x2 – 9
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
1
2
3
4
5
6
7
8
9 10
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
Algebra 1 Notes: Chapter 9
- 25 Notes #30: Solving Radical Equations with Quadratics (Section 10.4)
Solving Radical Equations:
 Isolate the _________________
 ______________ both sides. If one side is a binomial, be sure to use ___________ to
square it.
 Get all terms to one side to = 0
 Solve the quadratic using: factoring, quadratic formula, or completing the square.
 Check your solution by ___________________ into the original equation. Check for
extraneous roots.
1.) 2 x  3  1
2.) 4  x  2  5
3.)
x2  x
4.)
35  2x  x
5.)
6x  9  x
6.)
11x  28  x
Algebra 1 Notes: Chapter 9
- 26 7.)
2x  7  x  2
8.)
3x  2  4  x
9.) Graph the quadratic. Find the requested
information:
f(x)= -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
1
2
3
4
5
6
7
8
9 10
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________