Transcript Document
6.1 Solving Quadratic Equations by Graphing Need Graph Paper!!!
Objective: 1) 2) 3) To write functions in quadratic form To graph quadratic functions To solve quadratic equations by graphing
• • • • • • •
Vocabulary Quadratic function Quadratic term Linear term Constant term Parabola-
bx c ax
2
f
(
x
)
ax
2
bx
c
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
Identify the quadratic term, the linear term, and the constant term.
1 1)
g
(
x
) 5
x
2 7
x
2 2)
f
(
n
) 3
n
2 4 3)
f
(
x
) (
x
3 ) 2
1) Use the related graph of each equation to determine its solutions and find the minimum or maximum point.
2
x
2 2
x
4 0 2)
x
2 10
x
25 0
Graph each function. Name the vertex and axis of symmetry.
3)
h
(
x
)
x
2 4
4) Graph each function. Name the vertex and axis of symmetry.
x
2 7
x
7
5) Solve by graphing. (Find the roots) 4
x
2 8
x
5
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) 0 48
t
16
t
2 2)
x
2 4 3) 3
x
2 13
x
10
Solve by using he zero product property.
4) (3y – 5)(2y + 7) = 0 5) x(x – 1) = 0 6) 3
c
2 5
c
Solve by using he zero product property.
7)
y
2
y
30 0 8) 18
r
3 16
r
34
r
2
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) 4
x
2 5
x
21 0 1) 2) 3)
Steps
The quadratic and linear term must be on one side of the equation and the constant must be on the other side.
The quadratic term must have a coefficient of 1.
Find c by taking half of the linear term and squaring it.
Solve by completing the square.
2) 2
x
2 7
x
12 0 1) 2) 3)
Steps
The quadratic and linear term must be on one side of the equation and the constant must be on the other side.
The quadratic term must have a coefficient of 1.
Find c by taking half of the linear term and squaring it.
Solve by completing the square.
3)
x
2 2
x
120 0 1) 2) 3)
Steps
The quadratic and linear term must be on one side of the equation and the constant must be on the other side.
The quadratic term must have a coefficient of 1.
Find c by taking half of the linear term and squaring it.
4) The distance d that an object tra vels can be calculated when the initial speed
v i
, elapsed time t, and the rate of constant accelerati on a are known.
A formula that relates these factors is
d
(
t
)
v i t
1 2
at
2 .
If a motorcycle has an initial speed of 30 m/s and a constant accelerati on of 6
m
/
s
2 , how much time will it take to travel 200 meters?
5) A ball is thrown straight up with an initial velocity of 57.7
feet per second.
The height of the ball t seconds after it is thrown is given by the formula
h
(
t
) 57 .
7
t
16
t
2 .
a.) What is the height of the ball after 1.3
seconds?
b.) What is its maximum height?
c.) After how many seconds will it return to the ground
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 formula 2) To use the discriminant to determine the nature of the roots of quadratic equations
Use quadratic formula to solve each equation.
(1.) 3
x
2 4
x
4 b x b 2 2 a 4 ac
Use quadratic formula to solve each equation.
(2.)
x
2 7
x
18 b x b 2 2 a 4 ac
Examples
1 2 3 discrimina nt
b
2 4
ac
Value of
b
2 4
ac
Greater than zero
Discriminant a Perfect Square?
Yes
Nature of Roots
2 real, rational #’s Greater than zero Yes 2 real, Irrational #’s Less than zero na 2 imaginary #’s 4 Zero na 1 real #
Find the value of the discriminant for each quadratic equation. Then 3) describe the nature of the roots.
x
2 8
x
16 4) 5
x
2 42 0
Find the value of the discriminant for each quadratic equation. Then 5) describe the nature of the roots.
x
2 5
x
50 0 6) 2
x
2 9
x
8 0
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 equations 2) 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.
Sum of Roots
r
1
r
2
a b
Quadratic Equation
ax
2
bx
c
0 Product of Roots
r
1 (
r
2 )
c a
(1) Write a quadratic equation that has roots ¾ and –12/5.
(Denominators must be the same) Sum of Roots
r
1
r
2
a b
Product of Roots
r
1 (
r
2 )
c a
(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.
i
3
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
y
a
(
x
h
) 2
k
2) To determine the equation of a parabola by using points on its graph.
1) Write the equation in the form
y
a
(
x
h
) 2
k
. Then name the vertex, axis of symmetry, and the direction of the opening.
f
(
x
) 3
x
2 18
x
11 2)
f
(
x
) 4 (
x
3 ) 2 1
Write the equation in the form
y
a
(
x
h
) 2
k
. Then name the 3) vertex, axis of symmetry, and the direction of the opening.
f
(
x
) 1 2
x
2 5
x
27 2 4)
f
(
x
) 4
x
2 24
x
5) Write the equation for each parabola and then state the domain and range in interval notation.
(1, 4) (3, 4) (2, 0)
6) Write the equation for each parabola and then state the domain and range in interval notation.
(-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)
9) Graph each function in the form
y
a
(
x
h
) 2
k
. Then name the vertex, axis of symmetry, and the direction of the opening. Write the domain and range in interval notation.
f
(
x
)
x
2 6
x
2
10) Graph each function in the form
y
a
(
x
h
) 2
k
. Then name the vertex, axis of symmetry, and the direction of the opening. Write the domain and range in interval notation.
f
(
x
) 9
x
2 18
x
6
Assignment 6.6
Page 373 (19-49 odd), 58, 62, 63, 64
6.7 Graphing and Solving Quadratic Inequalities
Objective: 1) To graph quadratic inequalities 2) To solve quadratic inequalities in one variable.
Use the General Form to graph parabolas (Complete the Square) 1)
y
2
x
2 8
x
9
Vertex:
( , )
Axis of Symmetry:
x=
Opening: Left Point and Right Point (x)
Use the General Form to graph parabolas (Complete the Square) 2)
y
x
2 4
Vertex:
( , )
Axis of Symmetry:
x=
Opening: Left Point and Right Point (x)
• Solve each inequality.
(3)
x
2 5
x
4 0 (1) Solve of x (2) Plot x’s on # line (3) Test point in each region (yes or no) (4) Write inequality (5) Write answer in interval notation
• Solve each inequality.
(4) 5
x
2 27
x
10 0 (1) Solve of x (2) Plot x’s on # line (3) Test point in each region (yes or no) (4) Write inequality (5) Write answer in interval notation
• Solve each inequality.
(5) (x – 1)(x + 4) (x – 3) > 0 (1) Solve of x (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 Review Exploring 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