Chapter 5 Quadratic
Functions
5.1 Graphing Quadratic
Functions
Vocabulary
• Quadratic function -
y  ax  bx  c, where a  0
2
• Parabola – graph of quadratic function (U-shaped)
• Vertex – lowest or highest point of quadratic function
• Axis of symmetry – vertical line through the vertex
Vocabulary
• Standard form
y  ax  bx  c
• Vertex form
y  a( x  h)  k
• Intercept form
y  a ( x  p)( x  q )
2
2
Standard form y  ax  bx  c
2
• Graph opens up if a > 0 and opens down if a < 0. The
2
parabola is wider than y  x if a  1 and narrower
2
than y  x if a  1
• The x-coordinate of the vertex is

b
2a
• The axis of symmetry is the vertical line
x   2ba
Graph y  2 x  8x  6
2
y  x  4x  1
2
Vertex Form y  a( x  h)  k
2
• The vertex is (h,k)
• Axis of symmetry is x = h
Graph y   ( x  3)  4
1
2
2
y  2( x  1)  4
2
Intercept Form y  a ( x  p)( x  q )
• X-intercepts are p and q.
• Axis of symmetry is halfway between (p, 0)
and (q,0).
Graph y  ( x  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  ( x  2)  3
1
2
2
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?
2
9
y   x( x  6)
5.7 Graphing and Solving
Quadratic Inequalities
Quadratic Inequalities
y  ax  bx  c
y  ax  bx  c
y  ax  bx  c
y  ax  bx  c
2
2
2
2
Graphing Quadratic Inequalities
1. Draw the parabola
dashed line if <, >
y  ax  bx  c
2
solid line if , 
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  2x  3
2
y  2 x  5x  3
2
y  x 4
2
y  x  x  2
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  bx  c  ( x  m)( x  n)
2
 x  (m  n) x  mn
2
Factor
x  2 x  48
2
x  bx  c
2
Factor
4y  4y  3
2
ax  bx  c
2
Difference of Two Squares
a  b  (a  b)(a  b)
2
2
x  9  ( x  3)( x  3)
2
Perfect Square Trinomial
a  2ab  b  (a  b)
2
x  12 x  36  ( x  6)
2
2
2
2
Perfect Square Trinomial
a  2ab  b  (a  b)
2
2
x  8 x  16  ( x  4)
2
2
2
Factor
9 x  36
2
Factor
1.
2.
4 x  16 x  36
2
x  6x  9
2
Factor Monomials First
4 x  20x  24
2
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  12 x  4  0
2
Solve the Quadratic
3x  6  x  10
2
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
Vocabulary
• Square root – 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  a  b
• Quotient property
a
b

a
b
Simplify the Expression
1.
500 
2.
6 8 
3.
3 12  6 
Simplify the Expression
1.
25
3

2.
2
11

Solve
3x  4  23
2
Solve
1.
3( x  2)  21
2. 1
5
2
( x  4)  6
2
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?
2
0
h  16t  h
5.4 Complex Numbers
Vocabulary
• Imaginary unit (i)
i  1
• Complex number
a  bi
i  1
2
– a is real part, bi is imaginary part
• Standard form
a  bi
• Imaginary number
a  bi , b  0
Vocabulary
• Pure imaginary number
a  bi , a  0 and b  0
• Complex plane
– Horizontal axis is real axis, vertical axis is imaginary
axis
• Complex conjugates
a  bi and a  bi
• Absolute value
– Distance from the origin in the complex plane
Square root of a negative number
1. If r is a positive real number, then
r  i r
5  i 5
2. Following step 1:
(i 5)  i  5  5
2
2
Solve
2 x  26  10
2
Solve
 ( x  1)  5
1
2
2
Plotting the Complex Numbers
a.
4  i
b.
1  3i
c.
5
Write in Standard Form
( 1  2i )  (3  3i )
Write in Standard Form
(2  3i )  (3  7i )
Write in Standard Form
1.
2i  (3  i )  (2  3i )
2.
i (3  i )
Write in Standard Form
1.
(2  3i )( 6  2i )
2.
(1  2i )(1  2i )
Write in Standard Form
5 3i
1 2 i
Write in Standard Form
2  7i
1 i
Absolute Value of a Complex Number
• Complex number:
z  a  bi
• Find absolute value using:
z  a b
2
2
Find the Absolute Value
2  5i
Find the Absolute Value
1.
5  3i
2.
6i
5.5 Completing the Square
Vocabulary
• Completing the Square – writing an equation
2
of the form: x  bx
as a square of a binomial
x  bx  ( )  ( x  )
2
b 2
2
b 2
2
Goal:
• Find the value of c that makes a perfect square
trinomial.
• Ex:
x  6x  9  ( x  3)
2
2
Find c, then write the expression as
the square of a binomial.
x  7x  c
2
Find c, then write the expression as
the square of a binomial.
x  11x  c
2
Solve by Completing the Square
x  10 x  3  0
2
Solve by Completing the Square
x  4x  1  0
2
Solve by Completing the Square
3x  6 x  12  0
2
Solve by Completing the Square
5x  10x  30  0
2
On dry asphalt, the formula for a car’s stopping
distance is given by d  0.05s 2  11
.s
What speed are you driving if you need 100 feet to
stop before an intersection?
Writing in Vertex form
• Standard form:
• Vertex form:
y  ax  bx  c
2
b g
y  a xh k
2
Write in vertex form.
Then find the vertex.
y  x  6x  16
2
Write in vertex form.
Then find the vertex.
y  x  3x  3
2
5.6 The Quadratic Formula
and the Discriminant
Vocabulary
• Quadratic formula – solves any quadratic
equation for x.
• Discriminant – determines the number and
type of solutions
Quadratic Formula
ax  bx  c  0
2
x
 b  b  4 ac
2a
2
Solve
1.
2x  x  5
2.
3x  8 x  35
2
2
Solve
1.
12 x  5  2 x  13
2.
2 x  2 x  3
2
2
Discriminant
x
 b  b 2  4 ac
2a
Discriminant
Number and Type of Solutions
• Two real solutions if
• One real solution if
b
b
2
 4 ac  0
2
 4 ac  0
2
 4 ac  0
• Two imaginary solutions if b
Find the discriminant, and give the
number and type of solutions.
9 x  6x  1  0
2
Find the discriminant, and give the
number and type of solutions.
1.
9 x  6x  4  0
2.
5x  3x  1  0
2
2
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  16t  v0t  h0
2
5.7 Solving Quadratic
Inequalities cont.
Solving Inequalities by Graphing
To solve ax  bx  c  0, graph y  ax  bx  c
2
2
and identify the x - values that lie below the x - axis.
To solve ax  bx  c  0, graph y  ax  bx  c
2
2
and identify the x - values that lie above the x - axis.
Solve the Inequality
2
x  5x  6  0
Solve the Inequality
x  x2  0
2
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.
Solve Algebraically
2x  x  3
2
Solve Algebraically
3x  11x  4
2
5.8 Modeling with Quadratic
Functions
Writing Quadratics in Vertex Form
• Vertex form:
b g
y  a xh k
2
• 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  a ( x  p)( x  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)