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6.1 Solving Quadratic Equations by Graphing
Need Graph Paper!!!Objective:1) To write functions in quadratic form2) To graph quadratic functions3) To solve quadratic equations by graphing
• Vocabulary
• Quadratic function-
• Quadratic term-
• Linear term-
• Constant term-
• Parabola- the graph of a quadratic function
• Axis of Symmetry- a line that makes the parabola symmetric
• Vertex- the minimum or maximum point of the parabola
• Zeros- the x-intercepts of the parabola
cbxaxxf 2)(2ax
bx
c
Identify the quadratic term, the linear term, and the constant term.
1) 2) 3)275)( 2 xxxg 4
3
1)( 2 nnf 2)3()( xxf
Use the related graph of each equation to determine its solutions and find the minimum or maximum point.
1) 2)0422 2 xx 025102 xx
Graph each function. Name the vertex and axis of symmetry.
3)4)( 2 xxh
Graph each function. Name the vertex and axis of symmetry.
4)2( ) 7 7g x x x
Solve by graphing. (Find the roots)
5)584 2 xx
Solve by graphing. (Find the roots)
5) (3x + 4)(2x + 7) = 0
Assignment 6.1
Page 339 (17-29 odd), (35- 41 odd), 49, 50, 51, 52
6.2 Solving Quadratic Equations by Factoring
Objective:
1) To solve problems by factoring
Solve by using he zero product property.
1) 2) 3) 20 48 16t t 42 x 10133 2 xx
Solve by using he zero product property.
4) (3y – 5)(2y + 7) = 0 5) x(x – 1) = 0 6) cc 53 2
Solve by using he zero product property.
7) 8) 0302 yy 23 341618 rrr
Assignment 6.2
Page 344 (11-33 odd), 41, 43, 44, 45, 46
6.3 Completing the Square
Objective:
1) To solve quadratic equations by completing the square
Solve by completing the square.
1)
Steps
1) The quadratic and linear term must be on one side of the equation and the constant must be on the other side.
2) The quadratic term must have a coefficient of 1.
3) Find c by taking half of the linear term and squaring it.
02154 2 xx
Solve by completing the square.
2)
Steps
1) The quadratic and linear term must be on one side of the equation and the constant must be on the other side.
2) The quadratic term must have a coefficient of 1.
3) Find c by taking half of the linear term and squaring it.
01272 2 xx
Solve by completing the square.
3)
Steps
1) The quadratic and linear term must be on one side of the equation and the constant must be on the other side.
2) The quadratic term must have a coefficient of 1.
3) Find c by taking half of the linear term and squaring it.
012022 xx
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Assignment 6.3
Page 351 (21-35 odd) 41, 43, 44, 46, 47
6.4 The Quadratic Formula and the Discriminant
Objective:1) To solve quadratic equations by using the
quadratic formula2) To use the discriminant to determine the
nature of the roots of quadratic equations
Use quadratic formula to solve each equation.
(1.) 443 2 xxx
bb2 4ac
2a
Use quadratic formula to solve each equation.
(2.) 1872 xxx
bb2 4ac
2a
Examples Value of Discriminant a Perfect Square?
Nature of Roots
1 Greater than zero
Yes 2 real, rational #’s
2 Greater than zero
Yes 2 real,
Irrational #’s
3 Less than zero na 2 imaginary #’s
4 Zero na 1 real #
acb 42
acb 4ntdiscrimina 2
Find the value of the discriminant for each quadratic equation. Then describe the nature of the roots.
3) 4)1682 xx 0425 2 x
Find the value of the discriminant for each quadratic equation. Then describe the nature of the roots.
5) 6)05052 xx 0892 2 xx
Assignment 6.4
Page 357 (17-29 odd), 34, 35, 36, 37, 38
6.5 Sum and Product of Roots
Objective:1) To find the sum and product of the roots of
quadratic equations2) To find a quadratic equation to fit a given
condition
• Quadratic equations can have up to 2 real roots (answers).
• The sum and the product of these roots can be used to write a quadratic equation.
Quadratic Equation
Sum of Roots Product of Roots02 cbxax
a
brr
21
a
crr )( 21
(1) Write a quadratic equation that has roots ¾ and –12/5.
(Denominators must be the same)
Sum of Roots Product of Roots
a
brr
21
a
crr )( 21
(2) Write a quadratic equation that has roots 3/2 and 1/4.
(Show the easier way to solve these problems)
(3) Write a quadratic equation that has roots 7 – 3i and 7 + 3i.
(4) Write a quadratic equation that has roots 6 and -9.
(5) Write a quadratic equation that has roots . 35 and 35 ii
Assignment 6.5
Page 363 (17-26), For (29-37 odd) solve each equation by using factoring, completing the square, or quadratic formula. Use each method at least once. 47, 48, 49, 51, 52
6.6 Analyzing Graphs of Quadratic Functions
Need Graph Paper!!!Objective:1) To graph quadratic functions of the form
2) To determine the equation of a parabola by using points on its graph.
khxay 2)(
Write the equation in the form . Then name the vertex, axis of symmetry, and the direction of the opening.
1) 2)
khxay 2)(
11183)( 2 xxxf 1)3(4)( 2 xxf
Write the equation in the form . Then name the vertex, axis of symmetry, and the direction of the opening.
3) 4)
khxay 2)(
2
275
2
1)( 2 xxxf xxxf 244)( 2
Write the equation for each parabola and then state the domain and range in interval notation.
5)
(1, 4) (3, 4)
(2, 0)
Write the equation for each parabola and then state the domain and range in interval notation.
6)
(-5, 2)
(-3, 6)
(-1, 2)
Write the equation for the parabola that passes through the given points.
7) (0, 0), (2, 6), (-1, 3) 8) (1, 0), (3, 38), (-2, 48)
Graph each function in the form . Then name the vertex, axis of symmetry, and the direction of the opening. Write the domain and range in interval notation.
9)
khxay 2)(
26)( 2 xxxf
Graph each function in the form . Then name the vertex, axis of symmetry, and the direction of the opening. Write the domain and range in interval notation.
10)
khxay 2)(
6189)( 2 xxxf
Assignment 6.6
Page 373 (19-49 odd), 58, 62, 63, 64
6.7 Graphing and Solving Quadratic Inequalities
Objective:1) To graph quadratic inequalities2) To solve quadratic inequalities in one
variable.
Use the General Form to graph parabolas (Complete the Square)
1)
Vertex: ( , )
Axis of Symmetry: x=
Opening:
Left Point and Right Point (x)
982 2 xxy
Use the General Form to graph parabolas (Complete the Square)
2)
Vertex: ( , )
Axis of Symmetry: x=
Opening:
Left Point and Right Point (x)
42 xy
• Solve each inequality. (1) Solve of x
(3) (2) Plot x’s on # line
(3) Test point in each region
(yes or no)
(4) Write inequality
(5) Write answer in interval notation
0452 xx
• Solve each inequality. (1) Solve of x
(4) (2) Plot x’s on # line
(3) Test point in each region
(yes or no)
(4) Write inequality
(5) Write answer in interval notation
010275 2 xx
• Solve each inequality. (1) Solve of x
(5) (x – 1)(x + 4) (x – 3) > 0 (2) Plot x’s on # line
(3) Test point in each region
(yes or no)
(4) Write inequality
(5) Write answer in interval notation
Assignment 6.7
Page 382 (27-53 odd), 63, 65, 66, 67, 68, 69, 70, 71
Unit 6 ReviewExploring Quadratic Functions and
Inequalities
• Unit 6 Test is worth 100 points
• Covers sections 6.1 – 6.7
• Study notes and hw
• Unit 6 Test Review
• Page 400 (11-53 odd)
• Page 357 (19, 23, 27)
• Page 382 (39, 47, 51)
• Page 352 (41)- worth 18 points on test
• Items on the Test• Quadratic function• Quadratic term• Linear term• Constant term• Parabola• Axis of Symmetry• Vertex• Zeros• Completing the Square• Quadratic Formula• Discriminant• Sum and Product of
Roots
• Domain• Range• Interval Notation• Intercepts• Quadratic Inequalities
