Transcript Chapter 5 Quadratic Functions
Chapter 5 Quadratic Functions
5.1 Graphing Quadratic Functions
• Quadratic function Vocabulary -
y
ax
2
bx
c
, where
a
0 • Parabola – graph of quadratic function (U-shaped) • Vertex – lowest or highest point of quadratic function • Axis of symmetry – vertical line through the vertex
• Standard form Vocabulary
y
ax
2
bx
c
• Vertex form
y
• Intercept form
y
h
) 2
k
q
)
Standard form
y
ax
2
bx
c
• Graph opens up if a > 0 and opens down if a < 0. The parabola is wider than than
y
x
2 if
a
1
y
x
2 if
a
1 and narrower • The x-coordinate of the vertex is
b
2
a
• The axis of symmetry is the vertical line
x
b
2
a
Graph
y
2
x
2 8
x
6
y
4
x
1
Vertex Form
y
• The vertex is (h,k) • Axis of symmetry is x = h
h
) 2
k
Graph
y
1 2 (
x
3 ) 2 4
y
2 (
x
1 ) 2 4
Intercept Form
y
• X-intercepts are p and q.
• Axis of symmetry is halfway between (p, 0) and (q,0).
q
)
Graph
y
2 )(
x
4 )
y
2 (
x
3 )(
x
1 )
Writing Quadratics in Standard Form 1. Use FOIL to expand the equation.
2. Combine like terms.
3. Write answer in decreasing order of degree (exponent).
Write in Standard Form
y
3 (
x
1 )(
x
5 )
Write in Standard Form
y
1 2 (
x
2 ) 2 3
The distance a woodland jumping mouse can hop is modeled by the following equation where x and y are measured in feet. How high can it hop? How far can it hop?
y
2 9 6 )
5.7 Graphing and Solving Quadratic Inequalities
Quadratic Inequalities
y
ax
2
bx
c y
ax
2
bx
c y
ax
2
bx
c y
ax
2
bx
c
Graphing Quadratic Inequalities 1. Draw the parabola dashed line if <, >
y
ax
2
bx
solid line if
c
2. Choose a point (x,y) inside the parabola and check if it’s a solution.
3. If it is a solution, shade inside the parabola. If it’s not a solution, shade the region outside the parabola.
y
x
2 2
x
3
y
2
x
2 5
x
3
y
x
2 4
y x
2
5.2 Solving Quadratic Equations by Factoring
Vocabulary • Binomial – contain two terms • Trinomial – contain three terms • Monomial – contain one term • Factoring – write a trinomial as a product of binomials (reverse FOIL) • Zeros – x-intercepts
Use FOIL to Simplify (
x
3 )(
x
5 )
Factoring
x
2
bx
x
2
( (
x
mn
n
)
Factor
x
2 2
x
48
x
2
bx
c
Factor
ax
2
bx
c
4
y
2 4
y
3
Difference of Two Squares
a
2
b
2 ( )( )
x
2 (
x
3 )(
x
3 )
Perfect Square Trinomial
a
2 2 2 (
a b
) 2
x
2 12
x
36 (
x
6 ) 2
Perfect Square Trinomial
a
2 2 2 (
a b
) 2
x
2 8
x
16 (
x
4 ) 2
9
x
2 36 Factor
1.
4
x
2 Factor 16
x
36 2.
x
2 6
x
9
Factor Monomials First 4
x
2 20
x
24
Zero Product Property • Let A and B be real numbers or algebraic expressions. If AB = 0, then A = 0 or B = 0.
Solve the Quadratic 9
x
2 12
x
0
Solve the Quadratic 3
x x
2 10
An artist is making a rectangular painting. She wants the length of the painting to be 4 feet more than twice the width of the painting. The area of the painting must be 30 square feet. What should the dimensions of the painting be?
5.3 Solving Quadratic Equations by Finding Square Roots
• Square root Vocabulary – has two solutions,
x
• Radical sign • Radicand – number beneath radical sign • Radical -
x
• Rationalizing the denominator – multiply both the numerator and denominator by the radical in the denominator in order to cancel it out
Properties of Square Roots • Product property
ab
• Quotient property
a b
a b a
b
1.
Simplify the Expression 500 2.
6 8 3.
3 12 6
1.
2.
Simplify the Expression 25 3 2 11
3
x
2 Solve 23
1.
Solve 3 (
x
2 ) 2
21 2.
1 5 (
x
4 ) 2 6
How long would it take an object dropped from a 550 foot tall tower to land on the roof of a 233 foot tall building?
h
16
t
2
h
0
5.4 Complex Numbers
Vocabulary • Imaginary unit
(i) i
1 • Complex number –
a
is real part,
bi a
bi
is imaginary part • Standard form
a
bi i
2 1 • Imaginary number 0
Vocabulary • Pure imaginary number , • Complex plane 0 and
b
0 – Horizontal axis is real axis, vertical axis is imaginary axis • Complex conjugates
a
bi
and
a
• Absolute value
bi
– Distance from the origin in the complex plane
Square root of a negative number 1. If
r
is a positive real number, then
i r i
5 2. Following step 1: (
i
5 ) 2 2 5 5
Solve 2
x
2 26 10
Solve 1 2 (
x
1 ) 2 5
a.
Plotting the Complex Numbers
i
b.
1 3
i
c.
5
Write in Standard Form ( 1 2
i
)
i
)
Write in Standard Form (
i
3 7
i
)
1.
2
i
Write in Standard Form
i i
) 2.
i
( 3
i
)
Write in Standard Form 1.
(
i
)( 6 2
i
) 2.
(
i
)(
i
)
Write in Standard Form 5 3 1 2
i i
Write in Standard Form 2 7 1
i i
Absolute Value of a Complex Number • Complex number:
z bi
• Find absolute value using:
z
a
2
b
2
Find the Absolute Value
i
Find the Absolute Value 1.
5 3
i
2.
6
i
5.5 Completing the Square
Vocabulary • Completing the Square of the form:
x
2
bx
– writing an equation as a square of a binomial
x
2
bx
2 2 (
x
b
2 ) 2
Goal: • Find the value of
c
that makes a perfect square trinomial.
• Ex:
x
2 6
x
(
x
3 ) 2
Find
c
, then write the expression as the square of a binomial.
x
2 7
x
c
Find
c
, then write the expression as the square of a binomial.
x
2 11
x
c
Solve by Completing the Square
x
2 10
x
0
Solve by Completing the Square
x
2 4
x
0
Solve by Completing the Square 3
x
2 6
x
12 0
Solve by Completing the Square 5
x
2 10
x
30 0
On dry asphalt, the formula for a car’s stopping distance is given by
d
.
s
2 .
s
What speed are you driving if you need 100 feet to stop before an intersection?
Writing in Vertex form • Standard form:
y
ax
2
bx
c
• Vertex form:
y
h
k
Write in vertex form. Then find the vertex.
y
x
2 6
x
16
Write in vertex form.
Then find the vertex.
y
x
2 3
x
3
5.6 The Quadratic Formula and the Discriminant
Vocabulary • Quadratic formula equation for x.
– solves any quadratic • Discriminant – determines the number and type of solutions
Quadratic Formula
ax
2
bx
0
x
b
2 4
ac
2
a
1.
2
x
2 Solve 5 2.
3
x
2 8
x
35
1.
12
x
Solve 2
x
2 13 2.
2
x
2 3
x
Discriminant
b
2 4
ac
2
a
Discriminant
Number and Type of Solutions • Two real solutions if
b
2 4
ac
0 • One real solution if
b
2 4
ac
0 • Two imaginary solutions if
b
2 4
ac
0
Find the discriminant, and give the number and type of solutions.
9
x
2 6
x
0
Find the discriminant, and give the number and type of solutions.
1.
9
x
2 6
x
0 2.
5
x
2 3
x
0
A man is standing on a roof top 100 feet above ground level. He tosses a penny up into the air with a velocity of 5 ft./sec. The penny leaves the man’s hand at four feet above the roof. How long will it take the penny to fall to the ground? Use the model below:
h
16
t
2
v t
0
h
0
5.7 Solving Quadratic Inequalities cont.
Solving Inequalities by Graphing To solve
ax
2
bx
0 , graph
y
ax
2
bx
c
To solve
ax
2
bx
0 , graph
y
ax
2
bx
c
Solve the Inequality
x
2 5
x
0
Solve the Inequality
x
2 2 0
Solve the Inequality Algebraically 1. Solve the equation for all x – values.
2. Find the critical x – values.
3. Plot the critical x – values on a number line using solid or open dots where necessary.
4. Break the number line into three intervals.
5. Test an x – value in each interval.
2
x
2 Solve Algebraically 3
Solve Algebraically 3
x
2 11
x
4
5.8 Modeling with Quadratic Functions
Writing Quadratics in Vertex Form • Vertex form:
y
h
k
• Information needed: – Vertex – One additional point on the parabola
Write in Vertex Form
Write in Vertex Form
Write in Vertex Form • Vertex (1,3) • Point (-1,-1)
Writing Quadratics in Intercept Form • Intercept form:
y
q
) • Information needed: – X - intercepts – One additional point on the parabola
Write in Intercept form
Write in Intercept form
Write in Intercept form • x- intercepts -1, 2 • Point (0,-4)