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Algebra 2 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 6-1 Graphing Quadratic Functions
Lesson 6-2 Solving Quadratic Equations by Graphing
Lesson 6-3 Solving Quadratic Equations by Factoring
Lesson 6-4 Completing the Square
Lesson 6-5 The Quadratic Formula and the Discriminant
Lesson 6-6 Analyzing Graphs of Quadratic Functions
Lesson 6-7 Graphing and Solving Quadratic Inequalities
Example 1 Graph a Quadratic Function
Example 2 Axis of Symmetry, y-Intercept, and Vertex
Example 3 Maximum or Minimum Value
Example 4 Find a Maximum Value
Graph
by making a table of values.
First, choose integer values for x. Then, evaluate the
function for each x value. Graph the resulting coordinate
pairs and connect the points with a smooth curve.
Answer:
x
f (x)
(x, y)
–3
–2
–1
0
1
–1
–3
–3
–1
3
(–3, –1)
(–2, –3)
(–1, –3)
(0, –1)
(1, 3)
Graph
Answer:
by making a table of values.
x
f(x)
–2
–1
0
1
2
4
1
2
7
16
Consider the quadratic function
Find the y-intercept, the equation of the axis of
symmetry, and the x-coordinate of the vertex.
Begin by rearranging the terms of the function so that the
quadratic term is first, the linear term is second and the
constant term is last. Then identify a, b, and c.
So,
and
The y-intercept is 2.
You can find the equation of the axis of symmetry using
a and b.
Equation of the axis of symmetry
Simplify.
Answer: The y-intercept is 2. The equation of the axis
of symmetry is x = 2.Therefore, the
x-coordinate of the vertex is 2.
Make a table of values that includes the vertex.
Choose some values for x that are less than 2 and some
that are greater than 2. This ensures that points on either
side of the axis of symmetry are graphed.
Answer:
x
0
1
2
3
4
f(x)
2
–1
–2
–1
2
(x, f(x))
(0, 2)
(1, –1)
(2, –2)
(3, –1)
(4, 2)
Vertex
Use this information to graph the function.
Graph the vertex and the
y-intercept.
Then graph the points from
your table connecting them
with a smooth curve.
(0, 2)
As a check, draw the axis of
symmetry,
, as a
dashed line.
The graph of the function
should be symmetric
about this line.
Answer:
(2, –2)
Consider the quadratic function
a. Find the y-intercept, the equation of the axis of symmetry,
and the x-coordinate of the vertex.
Answer: y-intercept: 3; axis of symmetry:
x-coordinate: 3
b. Make a table of values that includes the vertex.
Answer:
x
f(x)
0
3
1
–2
2
–5
3
–6
4
–5
5
–2
c. Use this information to graph the function.
Answer:
Consider the function
Determine whether the function has a maximum or a
minimum value.
For this function,
Answer: Since
the graph opens down and the
function has a maximum value.
State the maximum or minimum value of the function.
The maximum value of this function is the y-coordinate of
the vertex.
The x-coordinate of the vertex is
Find the y-coordinate of the vertex by evaluating the
function for
Original function
Answer: The maximum value of the function is 4.
Consider the function
a. Determine whether the function has a maximum or a
minimum value.
Answer: minimum
b. State the maximum or minimum value of the function.
Answer: –5
Economics A souvenir shop sells about 200
coffee mugs each month for $6 each. The
shop owner estimates that for each $0.50
increase in the price, he will sell about 10
fewer coffee mugs per month.
How much should the owner charge for
each mug in order to maximize the monthly
income from their sales?
Words
The income is the number of mugs multiplied
by the price per mug.
Variables Let
Then
the number of $0.50 price increases.
the price per mug and
the number of mugs sold.
Let I(x) equal the income as a function of x.
The
income
Equation I(x)
is
the number
of mugs sold
times
the price
per mug.
=
Multiply.
Simplify.
Rewrite in
form.
I(x) is a quadratic function with
and
Since
the function has a maximum
value at the vertex of the graph. Use the formula to
find the x-coordinate of the vertex.
x-coordinate of the vertex
Formula for the
x-coordinate of
the vertex
Simplify.
This means that the shop should make 4 price increases
of $0.50 to maximize their income.
Answer: The mug price should be
What is the maximum monthly income the owner can
expect to make from these items?
To determine the maximum income, find the maximum
value of the function by evaluating
Income function
Use a calculator.
Answer: Thus, the maximum income is $1280.
Check Graph this function on a graphing calculator, and
use the CALC menu to confirm this solution.
Keystrokes:
2nd
[CALC] 4 0
ENTER
At the bottom of the display
are the coordinates of the
maximum point on the graph
of
The y value of these
coordinates is the maximum
value of the function, or 1280.
10
ENTER
ENTER
Economics A sports team sells about 100 coupon
books for $30 each during their annual fund-raiser.
They estimate that for each $0.50 decrease in the
price, they will sell about 10 more coupon books.
a. How much should they charge for each book in order
to maximize the income from their sales?
Answer: $17.50
b. What is the maximum monthly income the team can
expect to make from these items?
Answer: $6125
Example 1 Two Real Solutions
Example 2 One Real Solution
Example 3 No Real Solution
Example 4 Estimate Roots
Example 5 Write and Solve an Equation
Solve
by graphing.
Graph the related quadratic function
The equation of the axis of symmetry is
Make a table using x values around
Then graph each point.
x
f (x)
–1
0
0
–4
1
–6
2
–6
3
–4
4
0
From the table and the graph,
we can see that the zeroes of
the function are –1 and 4.
Answer: The solutions of the equation are –1 and 4.
Check Check the solutions by substituting each solution
into the original equation to see if it is satisfied.
Solve
Answer:
–3 and 1
by graphing.
Solve
by graphing.
Write the equation in
form.
Add 4 to each side.
Graph the related quadratic function
x
0
1
2
3
4
f(x)
4
1
0
1
4
Notice that the graph has
only one x-intercept, 2.
Answer: The equation’s only solution is 2.
Solve
Answer: 3
by graphing.
Number Theory Find two real numbers whose
sum is 4 and whose product is 5 or show that
no such numbers exist.
Explore Let
Then
Plan
one of the numbers.
Since the product of the two numbers is 5,
you know that
Original equation
Distributive Property
Add x2 and subtract 4x
from each side.
Solve You can solve
related function
x
f (x)
0
5
by graphing the
.
1
2
2
1
3
2
4
5
Notice that the graph has no
x-intercepts. This means that
the original equation has no
real solution.
Answer: It is not possible for two numbers to have a sum
of 4 and a product of 5.
Examine Try finding the product of several
numbers whose sum is 4.
Number Theory Find two real numbers whose sum
is 7 and whose product is 14 or show that no such
numbers exist.
Answer: no such numbers exist
Solve
by graphing. If exact roots cannot
be found, state the consecutive integers between which
the roots are located.
The equation of the axis of symmetry of the related
function is
x
f(x)
0
3
1
–2
2
–5
3
–6
4
–5
5
–2
6
3
The x-intercepts of the graph
are between 0 and 1 and
between 5 and 6.
Answer: One solution is between 0 and 1
and the other is between 5 and 6.
Solve
by graphing. If exact roots cannot
be found, state the consecutive integers between which
the roots are located.
Answer: between 0 and 1 and between 3 and 4
Royal Gorge Bridge The highest bridge in the United
States is the Royal Gorge Bridge in Colorado. The
deck of the bridge is 1053 feet above the river below.
Suppose a marble is dropped over the railing from a
height of 3 feet above the bridge deck. How long will it
take the marble to reach the surface of the water,
assuming there is no air resistance? Use the formula
where t is the time in seconds and h0
is the initial height above the water in feet.
We need to find t when
and
Original equation
Graph the related function
using
a graphing calculator. Adjust your window so that the
x-intercepts are visible.
Use the zero feature, 2nd [CALC], to find the positive
zero of the function, since time cannot be negative.
Use the arrow keys to locate a left bound for the zero
and press ENTER .
Then locate a right bound and press
ENTER
twice.
Answer: The positive zero of the function is
approximately 8. It should take about 8 seconds for
the marble to reach the surface of the water.
Hoover Dam One of the largest dams in the United
States is the Hoover Dam on the Colorado River,
which was built during the Great Depression. The dam
is 726.4 feet tall. Suppose a marble is dropped over
the railing from a height of 6 feet above the top of the
dam. How long will it take the marble to reach the
surface of the water, assuming there is no air
resistance? Use the formula
where t is the time in seconds and h0 is the initial
height above the water in feet.
Answer: about 7 seconds
Example 1 Two Roots
Example 2 Double Root
Example 3 Greatest Common Factor
Example 4 Write an Equation Given Roots
Solve
by factoring.
Original equation
Add 4x to each side.
Factor the binomial.
or
Zero Product Property
Solve the second equation.
Answer: The solution set is {0, –4}.
Check Substitute 0 and –4 in for x in the original equation.
Solve
by factoring.
Original equation
Subtract 5x and 2 from
each side.
Factor the trinomial.
or
Zero Product Property
Solve each equation.
Answer: The solution set is
Check each solution.
Solve each equation by factoring.
a.
Answer: {0, 3}
b.
Answer:
Solve
by factoring.
Original equation
Add 9 to each side.
Factor.
or
Zero Product Property
Solve each equation.
Answer: The solution set is {3}.
Check
The graph of the related function,
intersects the x-axis only once. Since the zero of the
function is 3, the solution of the related equation is 3.
Solve
Answer: {–5}
by factoring.
Multiple-Choice Test Item
What is the positive solution of the
equation
?
A –3
B 5
C 6
D 7
Read the Test Item
You are asked to find the positive solution of the given
quadratic equation. This implies that the equation also has
a solution that is not positive. Since a quadratic equation
can either have one, two, or no solutions, we should
expect this equation to have two solutions.
Solve the Test Item
Original equation
Factor.
Divide each side by 2.
Factor.
or
Zero Product Property
Solve each equation.
Both solutions, –3 and 7, are listed among the answer
choices. However, the question asks for the positive
solution, 7.
Answer: D
Multiple-Choice Test Item
What is the positive solution of the
equation
?
B –5
A 5
C 2
D 6
Answer: C
Write a quadratic equation with
and 6 as its
roots. Write the equation in the form
where a, b, and c are integers.
Write the pattern.
Replace p with
and q with 6.
Simplify.
Use FOIL.
Multiply each side by 3
so that b is an integer.
Answer: A quadratic equation with roots
and 6
and integral coefficients is
You can check this result by graphing the
related function.
Write a quadratic equation with
and 5 as its
roots. Write the equation in the form
where a, b, and c are integers.
Answer:
Example 1 Equation with Rational Roots
Example 2 Equation with Irrational Roots
Example 3 Complete the Square
Example 4 Solve an Equation by Completing the Square
Example 5 Equation with a  1
Example 6 Equation with Complex Solutions
Solve
Root Property.
by using the Square
Original equation
Factor the perfect
square trinomial.
Square Root Property
Subtract 7 from each side.
or
Write as two equations.
Solve each equation.
Answer: The solution set is {–15, 1}.
Solve
Root Property.
Answer: {3, 13}
by using the Square
Solve
Root Property.
by using the Square
Original equation
Factor the perfect
square trinomial.
Square Root Property
Add 5 to each side.
or
Write as two equations.
Use a calculator.
Answer: The exact solutions of this equation are
and
The approximate solutions are 1.5 and 8.5.
Check these results by finding and graphing the related
quadratic function.
Original equation
Subtract 12 from each side.
Related quadratic function
Check Use the ZERO function of a graphing
calculator. The approximate zeros of the related
function are 1.5 and 8.5.
Solve
Root Property.
Answer:
by using the Square
Find the value of c that makes
a perfect
square. Then write the trinomial as a perfect square.
Step 1
Find one half of 16.
Step 2
Square the result of Step 1.
Step 3
Add the result of Step 2
to
Answer: The trinomial
as
can be written
Find the value of c that makes
a perfect
square. Then write the trinomial as a perfect square.
Answer: 9; (x + 3)2
Solve
by completing the square.
Notice that
is not a perfect square.
Rewrite so the left side
is of the form
Since
add 4 to each side.
Write the left side as a
perfect square by factoring.
Square Root Property
Subtract 2 from each side.
or
Write as two equations.
Solve each equation.
Answer: The solution set is {–6, 2}.
Solve
Answer: {–6, 1}
by completing the square.
Solve
by completing the square.
Notice that
is not a perfect square.
Divide by the coefficient
of the quadratic term, 3.
Add
to each side.
Since
add
to each side.
Write the left side as a
perfect square by factoring.
Simplify the right side.
Square Root Property
Add
to each side.
or
Write as two
equations.
Solve each equation.
Answer: The solution set is
Solve
Answer:
by completing the square.
Solve
by completing the square.
Notice that
is not a perfect square.
Rewrite so the left side
is of the form
Since
add 1 to each side.
Write the left side as a
perfect square by factoring.
Square Root Property
Subtract 1 from each side.
Answer: The solution set is
Notice that these are imaginary solutions.
Check
A graph of the related function shows that the
equation has no real solutions since the graph
has no x-intercepts. Imaginary solutions must
be checked algebraically by substituting them in
the original equation.
Solve
Answer:
by completing the square.
Example 1 Two Rational Roots
Example 2 One Rational Root
Example 3 Irrational Roots
Example 4 Complex Roots
Example 5 Describe Roots
Solve
by using the Quadratic Formula.
First, write the equation in the form
and identify a, b, and c.
Then, substitute these values into the Quadratic Formula.
Quadratic Formula
Replace a with 1, b
with –8, and c with –33.
Simplify.
Simplify.
or
Write as two equations.
Simplify.
Answer: The solutions are 11 and –3.
Solve
Answer: 2, –15
by using the Quadratic Formula.
Solve
by using the Quadratic Formula.
Identify a, b, and c. Then, substitute these values into the
Quadratic Formula.
Quadratic Formula
Replace a with 1, b with
–34, and c with 289.
Simplify.
Answer: The solution is 17.
Check
A graph of the related function shows
that there is one solution at
Solve
Quadratic Formula.
Answer: 11
by using the
Solve
by using the Quadratic Formula.
Quadratic Formula
Replace a with 1, b with
–6, and c with 2.
Simplify.
or
Answer: The exact solutions are
and
The approximate solutions are 0.4 and 5.6.
Check Check these results by graphing the related
quadratic function,
Using the
ZERO function of a graphing calculator, the
approximate zeros of the related function are
–2.9 and 0.9.
Solve
Answer:
by using the Quadratic Formula.
or approximately 0.7 and 4.3
Solve
by using the Quadratic Formula.
Write the equation in the form
Now use the Quadratic Formula.
Quadratic Formula
Replace a with 1, b with
–6, and c with 13.
Simplify.
Simplify.
Answer: The solutions are the complex numbers
and
A graph of the function shows that the solutions are
complex, but it cannot help you find them.
Check To check complex solutions, you must substitute
them into the original equations. The check for
is shown below.
Original equation
Sum of a square;
Distributive Property
Simplify.
Solve
Answer:
by using the Quadratic Formula.
Find the value of the discriminant for
.
Then describe the number and type of roots for the
equation.
Answer: The discriminant is 0, so there is one rational root.
Find the value of the discriminant for
.
Then describe the number and type of roots for the
equation.
Answer: The discriminant is negative, so there are
two complex roots.
Find the value of the discriminant for
.
Then describe the number and type of roots for the
equation.
Answer: The discriminant is 80, which is not a perfect
square. Therefore, there are two irrational roots.
Find the value of the discriminant for
Then describe the number and type of roots for the
equation.
.
Answer: The discriminant is 81, which is a perfect square.
Therefore, there are two rational roots.
Find the value of the discriminant for each quadratic
equation. Then describe the number and type of roots
for the equation.
a.
Answer: 0; 1 rational root
b.
Answer: –24; 2 complex roots
c.
Answer: 5; 2 irrational roots
d.
Answer: 64; 2 rational roots
Example 1 Graph a Quadratic Function in Vertex Form
Example 2 Write y = x2 + bx + c in Vertex Form
Example 3 Write y = ax2 + bx + c in Vertex Form, a  1
Example 4 Write an Equation Given Points
Analyze
Then draw its graph.
h = 3 and k = 2
Answer: The vertex is at (h, k) or (3, 2) and the axis of
symmetry is
The graph has the same
shape as the graph of
but is translated
3 units right and 2 units up.
Now use this information to draw the graph.
Step 1
Plot the vertex, (3, 2).
Step 2
Draw the axis of
symmetry,
Step 3
Find and plot two points
on one side of the axis
of symmetry, such as
(2, 3) and (1, 6).
Step 4
Use symmetry to
complete the graph.
(1, 6)
(5, 6)
(2, 3)
(3, 2)
(4, 3)
Analyze
Then draw its graph.
Answer:
The vertex is at (–2, –4), and the axis of symmetry is
The graph has the same shape as the graph of
; it is translated 2 units left and 4 units down.
Write
the function.
in vertex form. Then analyze
Notice that
is not a perfect square.
Complete the square by
adding
Balance this addition
by subtracting 1.
Write
as a perfect square.
This function can be rewritten as
So,
and
Answer: The vertex is at (–1, 3), and the axis of symmetry
is
Since
the graph opens up and has the
same shape as
but is translated 1 unit left and 3
units up.
Write
the function.
in vertex form. Then analyze
Answer:
vertex: (–3, –4); axis of
symmetry:
opens up; the graph has the same
shape as the graph of
but it is translated 3 units
left and 4 units down.
Write
in vertex form. Then analyze
and graph the function.
Original equation
Group
and
factor, dividing by a.
Complete the square
by adding 1 inside the
parentheses.
Balance this addition
by subtracting –2(1).
Write
as a perfect square.
Answer: The vertex form is
So,
and
The vertex is at (–1, 4) and the axis of
symmetry is
Since
the graph opens down
and is narrower than
It is also translated 1 unit left
and 4 units up.
Now graph the function. Two points on the graph to the
right of
are (0, 2) and (0.5, –0.5). Use symmetry to
complete the graph.
Write
in vertex form. Then analyze
and graph the function.
Answer:
vertex: (–1, 7); axis of symmetry:
x = –1; opens down; the graph is narrower than the graph
of y = x2,and it is translated 1 unit left and 7 units up.
Write an equation for the parabola whose vertex is at
(1, 2) and passes through (3, 4).
The vertex of the parabola is at (1, 2) so
and
Since (3, 4) is a point on the graph of the parabola
,
and
Substitute these values into the vertex form of
the equation and solve for a.
Vertex form
Substitute 1 for h, 2 for k,
3 for x, and 4 for y.
Simplify.
Subtract 2 from each side.
Divide each side by 4.
Answer: The equation of the parabola in vertex form
is
Check A graph of
verifies that the
parabola passes through the point at (3, 4).
Write an equation for the parabola whose vertex is at
(2, 3) and passes through (–2, 1).
Answer:
Example 1 Graph a Quadratic Inequality
Example 2 Solve ax2 + bx + c  0
Example 3 Solve ax2 + bx + c  0
Example 4 Write an Inequality
Example 5 Solve a Quadratic Inequality
Graph
Step 1
Graph the related quadratic
equation,
Since the inequality symbol
is >, the parabola should be
dashed.
Graph
Step 2
Test a point inside the
parabola, such as (1, 2).
So, (1, 2) is a solution of the inequality.
(1, 2)
Graph
Step 3
Shade the region inside
the parabola.
(1, 2)
Graph
Answer:
Solve
by graphing.
The solution consists of the x values for which the graph
of the related quadratic function lies above the x-axis.
Begin by finding the roots of the related equation.
Related equation
Factor.
or
Zero Product Property
Solve each equation.
Sketch the graph of the parabola that has x-intercepts at
3 and 1. The graph lies above the x-axis to the left of
and to the right of
Answer: The solution set is
Solve
Answer:
by graphing.
Solve
by graphing.
This inequality can be rewritten as
The
solution consists of the x-values for which the graph of the
related quadratic equation lies on and above the
x-axis. Begin by finding roots of the related equation.
Related equation
Use the Quadratic
Formula.
Replace a with –2,
b with –6 and c with 1.
or
Simplify and write
as two equations.
Simplify.
Sketch the graph of the parabola that has x-intercepts
of –3.16 and 0.16. The graph should open down
since a < 0.
Answer:
The graph lies on and above the x-axis at
and
and between these two values. The solution set
of the inequality is approximately
Check Test one value of x less than –3.16, one
between –3.16 and 0.16, and one greater than 0.16
in the original inequality.
Solve by
Answer:
by graphing.
Sports The height of a ball above the ground after it
is thrown upwards at 40 feet per second can be
modeled by the function
where the
height h(x) is given in feet and the time x is in
seconds. At what time in its flight is the ball within
15 feet of the ground?
The function h(x) describes the height of the ball.
Therefore, you want to find values of x for which
Original inequality
Subtract 15 from
each side.
Graph the related function
graphing calculator.
using a
The zeros are about 0.46 and 2.04. The graph lies below
the x-axis when
or
Answer: Thus, the ball is within 15 feet of the ground
for the first 0.46 second of its flight and again after 2.04
seconds until the ball hits the ground at 2.5 seconds.
Sports The height of a ball above the ground after it is
thrown upwards at 28 feet per second can be modeled
by the function
where the height
h(x) is given in feet and the time x is given in seconds.
At what time in its flight is the ball within 10 feet of the
ground?
Answer: The ball is within 10 feet of the ground for the
first 0.5 second of its flight and again after 1.25 seconds
until the ball hits the ground.
Solve
algebraically.
First, solve the related equation
.
Related quadratic equation
Subtract 2 from each side.
Factor.
or
Zero Product Property
Solve each equation.
Plot –2 and 1 on a number line. Use closed circles since
these solutions are included. Notice that the number line is
separated into 3 intervals.
Test a value in each interval to see if it satisfies the
original inequality.
Answer: The solution set is
This is shown on the number line below.
Solve
Answer:
algebraically.
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