Chapter 8 Review Quadratic Functions § 8.3 Graphing Quadratic Equations in Two Variables.
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Transcript of Chapter 8 Review Quadratic Functions § 8.3 Graphing Quadratic Equations in Two Variables.
Chapter 8Review
Quadratic Functions
§ 8.3
Graphing Quadratic Equations in Two
Variables
Martin-Gay, Developmental Mathematics 3
We spent a lot of time graphing linear equations in chapter 3.
The graph of a quadratic equation is a parabola.
The highest point or lowest point on the parabola is the vertex.
Axis of symmetry is the line that runs through the vertex and through the middle of the parabola.
Graphs of Quadratic Equations
Martin-Gay, Developmental Mathematics 4
x
y
Graph y = 2x2 – 4.
x y
0 –4
1 –2
–1 –2
2 4
–2 4
(2, 4)(–2, 4)
(1, –2)(–1, – 2)
(0, –4)
Graphs of Quadratic Equations
Example
Martin-Gay, Developmental Mathematics 5
Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points.
To find x-intercepts of the parabola, let y = 0 and solve for x.
To find y-intercepts of the parabola, let x = 0 and solve for y.
Intercepts of the Parabola
Martin-Gay, Developmental Mathematics 6
If the quadratic equation is written in standard form, y = ax2 + bx + c,
1) the parabola opens up when a > 0 and opens down when a < 0.
2) the x-coordinate of the vertex is . a
b
2
To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y.
Characteristics of the Parabola
Martin-Gay, Developmental Mathematics 7
x
yGraph y = –2x2 + 4x + 5.
x y
1 7
2 5
0 5
3 –1
–1 –1
(3, –1)(–1, –1)
(2, 5)(0, 5)
(1, 7)Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is 1
)2(2
4
Graphs of Quadratic Equations
Example
Martin-Gay, Developmental Mathematics 8
The Graph of a Quadratic Function
The vertical line which passes through the vertex is called the Axis of Symmetry or the Axis
Recall that the equation of a vertical line is x =c
For some constant c
x coordinate of the vertex
The y coordinate of the vertex is
a
b
2
a
bac
2
4 2
The Axis of Symmetry is the x =a
b
2
Martin-Gay, Developmental Mathematics 9
The Quadratic function
Opens up when a>o
axis of symmetry
opens down a < 0
Vertex
Martin-Gay, Developmental Mathematics 10
Identify the Vertex and Axis of Symmetry of a Quadratic Function
Vertex =(x, y). thusVertex =
Axis of Symmetry: the line x =
Vertex is minimum point if parabola opens up Vertex is maximum point if parabola opens down
a
bf
a
b
2,
2
a
b
2
Martin-Gay, Developmental Mathematics 11
Identify the Vertex and Axis of Symmetry
Vertex x =
y =
Vertex = (-1, -3)Axis of Symmetry is x =
)3(2
6
xxxf 63)( 2
a
b
2
= = -1
1
2f
a
bf -3
a
b
2
= -1
Martin-Gay, Developmental Mathematics 12
The number of real solutions is at most two.
8.5 Quadratic Solutions
No solutions One solution Two solutions
Martin-Gay, Developmental Mathematics 13
Example f(x) = x2 - 4
Identifying Solutions
Solutions are -2 and 2.
Martin-Gay, Developmental Mathematics 14
Now you try this problem.
f(x) = 2x - x2
Solutions are 0 and 2.
Identifying Solutions
Martin-Gay, Developmental Mathematics 15
The graph of a quadratic equation is a parabola.
The roots or zeros are the x-intercepts.
The vertex is the maximum or minimum point.
All parabolas have an axis of symmetry.
Graphing Quadratic Equations
Martin-Gay, Developmental Mathematics 16
One method of graphing uses a table with
arbitrary
x-values.Graph y = x2 - 4x
Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2
Graphing Quadratic Equations
x y0 01 -32 -43 -34 0
Martin-Gay, Developmental Mathematics 17
Try this problem y = x2 - 2x - 8.
RootsVertexAxis of Symmetry
Graphing Quadratic Equations
x y-2-1134
Martin-Gay, Developmental Mathematics 18
8.6 – Solving Quadratic Equations by Factoring
A quadratic equation is written in the Standard Form, 2 0ax bx c where a, b, and c are real numbers and .0a
Examples: 2 7 12 0x x
23 4 15x x 7 0x x
(standard form)
Martin-Gay, Developmental Mathematics 19
Zero Factor Property: If a and b are real numbers and if , 0ab
Examples:
7 0x x
then or . 0a 0b
0x 7 0x 7x 0x
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 20
Zero Factor Property: If a and b are real numbers and if , 0ab
Examples: 10 3 6 0x x
then or . 0a 0b
10 0x 3 6 0x
10x 3 6x 2x
10 10 01 0x 63 66 0x 3 6
3 3
x
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 21
Solving Quadratic Equations: 1) Write the equation in standard form.
4) Solve each equation.
2) Factor the equation completely.
3) Set each factor equal to 0.
5) Check the solutions (in original equation).
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 22
2 3 18 0x x
6 0x 3 0x 3x
6x 3x
2 3 18x x
18 :Factors of1,18 2, 9 3, 6
26 3 16 8
36 18 18
18 18
213 3 83
9 9 18 18 18
6x 0
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 23
3 18x x 18x
2 3 18x x
218 13 18 8
324 54 18
270 18
221 23 11 8
441 63 18 378 18
3 18x
3 183 3x 21x
If the Zero Factor Property is not used, then the solutions will be incorrect
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 24
2 4 5x x
1 0x 5 0x
1 5 0x x
1x 5x
4 5x x
2 4 5 0x x
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 25
23 7 6x x 3 0x 3 2 0x
3 3 2 0x x
3x 2
3x
3 7 6x x
23 7 6 0x x 3 2x
6 :Factors of2, 31, 6
3:Factors of1, 3
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 26
29 24 16x x 29 24 16 0x x
3 4 0x 3 4 3 4 0x x
4
3x
3 4x
9 16and are perfect squares
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 27
32 18 0x x 2x
2 0x
2x
3x 3 0x 3 0x
3x 0x
2 9x 0
3x 3x 0
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 28
23 3 20 7 0x x x
3x
3 0x
7x 7 0x 3 1 0x
1
3x
3x 3 1x
3:Factors of 1, 3 7 :Factors of 1, 7
7x 0 3 1x
8.6 – Solving Quadratic Equations by Factoring
Martin-Gay, Developmental Mathematics 29
0
A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: 216 64.h t How long does it take for the diver to hit the surface of the water?
0 0
2 0t 2 0t 2t 2t seconds
216 64t 16 2 4t
16 2t 2t
8.6 – Quadratic Equations and Problem Solving
Martin-Gay, Developmental Mathematics 30
2x
The square of a number minus twice the number is 63. Find the number.
7x
7x
x is the number.
2 2 63 0x x
7 0x 9 0x
9x
2x 63
63:Factors of 1, 63 3, 21 7, 9
9x 0
8.6 – Quadratic Equations and Problem Solving
Martin-Gay, Developmental Mathematics 31
5 176w w
The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden?
11w The width is w.
11 0w 11w
The length is w+5.l w A
2 5 176w w 2 5 176 0w w
16 0w 16w
11w 11 5l 16l
feet
feet
176 :Factors of1,176 2, 88 4, 44
8, 22 11,16
16w 0
8.6 – Quadratic Equations and Problem Solving
Martin-Gay, Developmental Mathematics 32
x
Find two consecutive odd numbers whose product is 23 more than their sum?
Consecutive odd numbers: x
5x 5x 2 2 2 25x x x
2 25 0x 5x
5 0x 5 0x
5, 3 5, 7
5 2 3 5 2 7
2.x 2x 2x x 23
2 22 2 2 25xx x x x
2 25 2525x
2 25x
5x 0
8.6 – Quadratic Equations and Problem Solving
Martin-Gay, Developmental Mathematics 33
a x
The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs?
12a
.Pythagorean Th
22 27 13x x
5x
5
meters
7b x 13c
2 2 14 49 169x x x 22 14 120 0x x
22 7 60 0x x
2
5 0x 12 0x 12x
12 7b meters
2 2 2a b c
60 :Factors of 1, 60 2, 303, 20 4,15 5,12
5x 12x 0
6,10
8.6 – Quadratic Equations and Problem Solving
§ 8.7
Solving Quadratic Equations by the Square
Root Property
Martin-Gay, Developmental Mathematics 35
Square Root Property
We previously have used factoring to solve quadratic equations.
This chapter will introduce additional methods for solving quadratic equations.
Square Root PropertyIf b is a real number and a2 = b, then
ba
Martin-Gay, Developmental Mathematics 36
Solve x2 = 49
2x
Solve (y – 3)2 = 4
Solve 2x2 = 4
x2 = 2
749 x
y = 3 2
y = 1 or 5
243 y
Square Root Property
Example
Martin-Gay, Developmental Mathematics 37
Solve x2 + 4 = 0 x2 = 4
There is no real solution because the square root of 4 is not a real number.
Square Root Property
Example
Martin-Gay, Developmental Mathematics 38
Solve (x + 2)2 = 25
x = 2 ± 5
x = 2 + 5 or x = 2 – 5
x = 3 or x = 7
5252 x
Square Root Property
Example
Martin-Gay, Developmental Mathematics 39
Solve (3x – 17)2 = 28
72173 x
3
7217 x
7228 3x – 17 =
Square Root Property
Example
§ 8.8
Solving Quadratic Equations by Completing
the Square
Martin-Gay, Developmental Mathematics 41
In all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left.
Also, the constant on the left is the square of the constant on the right.
So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x2 term is 1, as in our previous examples).
Completing the Square
Martin-Gay, Developmental Mathematics 42
What constant term should be added to the following expressions to create a perfect square trinomial?
x2 – 10xadd 52 = 25
x2 + 16xadd 82 = 64
x2 – 7x
add 4
49
2
72
Completing the Square
Example
Martin-Gay, Developmental Mathematics 43
We now look at a method for solving quadratics that involves a technique called completing the square.
It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section.
Completing the Square
Example
Martin-Gay, Developmental Mathematics 44
Solving a Quadratic Equation by Completing a Square
1) If the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient.
2) Isolate all variable terms on one side of the equation.
3) Complete the square (half the coefficient of the x term squared, added to both sides of the equation).
4) Factor the resulting trinomial.
5) Use the square root property.
Completing the Square
Martin-Gay, Developmental Mathematics 45
Solve by completing the square.
y2 + 6y = 8y2 + 6y + 9 = 8 + 9
(y + 3)2 = 1
y = 3 ± 1
y = 4 or 2
y + 3 = ± = ± 11
Solving Equations
Example
Martin-Gay, Developmental Mathematics 46
Solve by completing the square.
y2 + y – 7 = 0
y2 + y = 7
y2 + y + ¼ = 7 + ¼
2
29
4
29
2
1y
2
291
2
29
2
1 y
(y + ½)2 = 429
Solving Equations
Example
Martin-Gay, Developmental Mathematics 47
Solve by completing the square.
2x2 + 14x – 1 = 0
2x2 + 14x = 1
x2 + 7x = ½
2
51
4
51
2
7x
2
517
2
51
2
7 x
x2 + 7x + = ½ + = 4
49
4
49
4
51
(x + )2 = 4
51
2
7
Solving Equations
Example
§ 8.9
Solving Quadratic Equations by the
Quadratic Formula
Martin-Gay, Developmental Mathematics 49
The Quadratic Formula
Another technique for solving quadratic equations is to use the quadratic formula.
The formula is derived from completing the square of a general quadratic equation.
Martin-Gay, Developmental Mathematics 50
A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions.
a
acbbx
2
42
The Quadratic Formula
Martin-Gay, Developmental Mathematics 51
Solve 11n2 – 9n = 1 by the quadratic formula.
11n2 – 9n – 1 = 0, so
a = 11, b = -9, c = -1
)11(2
)1)(11(4)9(9 2
n
22
44819
22
1259
22
559
The Quadratic Formula
Example
Martin-Gay, Developmental Mathematics 52
)1(2
)20)(1(4)8(8 2
x
2
80648
2
1448
2
128 20 4 or , 10 or 22 2
x2 + 8x – 20 = 0 (multiply both sides by 8)
a = 1, b = 8, c = 20
8
1
2
5Solve x2 + x – = 0 by the quadratic formula.
The Quadratic Formula
Example
Martin-Gay, Developmental Mathematics 53
Solve x(x + 6) = 30 by the quadratic formula.
x2 + 6x + 30 = 0
a = 1, b = 6, c = 30
)1(2
)30)(1(4)6(6 2
x
2
120366
2
846
So there is no real solution.
The Quadratic Formula
Example
Martin-Gay, Developmental Mathematics 54
The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.
The discriminant will take on a value that is positive, 0, or negative.
The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively.
The Discriminant
Martin-Gay, Developmental Mathematics 55
Use the discriminant to determine the number and type of solutions for the following equation.
5 – 4x + 12x2 = 0
a = 12, b = –4, and c = 5
b2 – 4ac = (–4)2 – 4(12)(5)
= 16 – 240
= –224
There are no real solutions.
The Discriminant
Example
Martin-Gay, Developmental Mathematics 56
Solving Quadratic Equations
Steps in Solving Quadratic Equations1) If the equation is in the form (ax+b)2 = c, use
the square root property to solve.
2) If not solved in step 1, write the equation in standard form.
3) Try to solve by factoring.
4) If you haven’t solved it yet, use the quadratic formula.
Martin-Gay, Developmental Mathematics 57
Solve 12x = 4x2 + 4.
0 = 4x2 – 12x + 4
0 = 4(x2 – 3x + 1)
Let a = 1, b = -3, c = 1
)1(2
)1)(1(4)3(3 2
x
2
493
2
53
Solving Equations
Example
Martin-Gay, Developmental Mathematics 58
Solve the following quadratic equation.
02
1
8
5 2 mm
0485 2 mm
0)2)(25( mm
02025 mm or
25
2 mm or
Solving Equations
Example
Martin-Gay, Developmental Mathematics 59
The Quadratic Formula
Solve for x by completing the square.
a
acbbx
2
42
Martin-Gay, Developmental Mathematics 60
Yes, you can remember this formula
Pop goes the Weaselhttp://www.youtube.com/watch?v=2lbABbfU6Zc&feature=related
Gilligan’s Islandhttp://www.youtube.com/watch?v=3CWTt9QFioY&feature=related
This one I can’t explainhttp://www.youtube.com/watch?v=haq6kpWdEMs&feature=related
Martin-Gay, Developmental Mathematics 61
How does it work
Equation:
1
5
3
0153 2
c
b
a
xx
a
acbbx
2
42
Martin-Gay, Developmental Mathematics 62
How does it work
Equation:
1
5
3
0153 2
c
b
a
xx
6
13
6
5
6
135
6
12255
32
13455 2
x
x
x
a
acbbx
2
42
Martin-Gay, Developmental Mathematics 63
The Discriminant
The number in the square root of the quadratic formula.
acb 42
12425
6145
0652
2
xxGiven
Martin-Gay, Developmental Mathematics 64
The Discriminant
The Discriminant can be negative, positive or zero
If the Discriminant is positive,
there are 2 real answers.
If the square root is not a prefect square
( for example ),
then there will be 2 irrational roots
( for example ).
25
52
Martin-Gay, Developmental Mathematics 65
The Discriminant
The Discriminant can be negative, positive or zero
If the Discriminant is positive,
there are 2 real answers.
If the Discriminant is zero,there is 1 real answer.
If the Discriminant is negative,there are 2 complex
answers.complex answer have i.
Martin-Gay, Developmental Mathematics 66
Solve using the Quadratic formula
3382 xx
Martin-Gay, Developmental Mathematics 67
Solve using the Quadratic formula
12
331488
0338
338
2
2
2
x
xx
xx
Martin-Gay, Developmental Mathematics 68
Solve using the Quadratic formula
32
6
2
148
112
22
2
1482
148
2
1968
12
331488
0338
338
2
2
2
x
x
x
x
xx
xx
Martin-Gay, Developmental Mathematics 69
Solve using the Quadratic formula
0289342 xx
Martin-Gay, Developmental Mathematics 70
Solve using the Quadratic formula
12
289143434
028934
2
2
x
xx
Martin-Gay, Developmental Mathematics 71
Solve using the Quadratic formula
172
34
2
034
2
1156115634
12
289143434
028934
2
2
x
x
x
xx
Martin-Gay, Developmental Mathematics 72
Solve using the Quadratic formula
0262 xx
Martin-Gay, Developmental Mathematics 73
Solve using the Quadratic formula
732
72
2
6
2
286
2
8366
12
21466
026
2
2
x
x
x
xx
Martin-Gay, Developmental Mathematics 74
Solve using the Quadratic formula
12
131466
0136
613
2
2
2
x
xx
xx
Martin-Gay, Developmental Mathematics 75
Solve using the Quadratic formula
2
166
2
52366
12
131466
0136
613
2
2
2
x
x
xx
xx
Martin-Gay, Developmental Mathematics 76
Solve using the Quadratic formula
ix
ii
x
x
x
xx
xx
23
2
4
2
6
2
46
2
166
2
52366
12
131466
0136
613
2
2
2
Martin-Gay, Developmental Mathematics 77
Describe the roots
Tell me the Discriminant and the type of roots 0962 xx
Martin-Gay, Developmental Mathematics 78
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
0962 xx
Martin-Gay, Developmental Mathematics 79
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
0962 xx
0532 xx
Martin-Gay, Developmental Mathematics 80
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
-11, Two complex roots
0962 xx
0532 xx
Martin-Gay, Developmental Mathematics 81
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
-11, Two complex roots
0962 xx
0532 xx
0482 xx
Martin-Gay, Developmental Mathematics 82
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
-11, Two complex roots
80, Two irrational roots
0962 xx
0532 xx
0482 xx