Transcript Interactive Chalkboard - Aurora R

Algebra 1 Interactive Chalkboard
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Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 10-1
Graphing Quadratic Functions
Lesson 10-2
Solving Quadratic Equations by Graphing
Lesson 10-3
Solving Quadratic Equations by
Completing the Square
Lesson 10-4
Solving Quadratic Equations by Using the
Quadratic Formula
Lesson 10-5
Exponential Functions
Lesson 10-6
Growth and Decay
Lesson 10-7
Geometric Sequences
Example 1 Graph Opens Upward
Example 2 Graph Opens Downward
Example 3 Vertex and Axis of Symmetry
Example 4 Match Equations and Graphs
Use a table of values to graph
Graph these ordered pairs and connect
them with a smooth curve.
Answer:
x
y
–2
–4
–1
0
0
–2
1
–2
2
0
3
4
Use a table of values to graph
Answer:
x
y
–3
6
–2
3
–1
2
0
3
1
6
Use a table of values to graph
Graph these ordered pairs and connect
them with a smooth curve.
Answer:
x
y
–2
–8
–1
0
0
4
1
4
2
0
3
–8
Use a table of values to graph
Answer:
x
y
–3
–5
–2
0
–1
3
0
4
1
3
2
0
3
–5
Consider the graph of
Write the equation of the axis of symmetry.
In
Equation for the axis of symmetry
of a parabola
and
Answer: The equation of the axis of symmetry is
Consider the graph of
Find the coordinates of the vertex.
Since the equation of the axis of symmetry is x = –2 and
the vertex lies on the axis, the x-coordinate for the vertex
is –2.
Original equation
Simplify.
Add.
Answer: The vertex is at (–2, 6).
Identify the vertex as a maximum or minimum.
Answer: Since the coefficient of the x2 term is
negative, the parabola opens downward and the
vertex is a maximum point.
(–2, 6)
Graph the function.
You can use the symmetry of
the parabola to help you draw
its graph. On a coordinate
plane, graph the vertex and
the axis of symmetry.
Choose a value for x other
than –2. For example, choose
–1 and find the y-coordinate
that satisfies the equation.
Original equation
Simplify.
(–2, 6)
Graph the function.
Graph (–1, 4).
Since the graph is symmetrical
about its axis of symmetry
x = –2, you can find another
point on the other side of the
axis of symmetry. The point
at (–1, 4) is 1 unit to the right
of the axis. Go 1 unit to the
left of the axis and plot the
point (–3, 4).
(–3, 4)
(–1, 4)
(–2, 6)
Graph the function.
Repeat this for several
other points.
(–3, 4)
(–1, 4)
Then sketch the parabola.
(–4, –2)
(0, –2)
Consider the graph of
a. Write the equation of the axis of symmetry.
Answer:
b. Find the coordinates of the vertex.
Answer: (1, –2)
c. Identify the vertex as a maximum or minimum.
Answer: Since the coefficient of the x2 term is
positive, the parabola opens upward and the
vertex is a minimum point.
Consider the graph of
d. Graph the function.
Answer:
Multiple-Choice Test Item
Which is the graph of
A
B
C
D
Read the Test Item
You are given a quadratic function, and you are asked to
choose the graph that corresponds to it.
Solve the Test Item
Find the axis of symmetry of the graph
Equation for the axis of symmetry
and
The axis of symmetry is –1. Look at the graphs. Since
only choices C and D have this as their axis of symmetry,
you can eliminate choices A and B. Since the coefficient
of the x2 term is negative, the graph opens downward.
Eliminate choice C.
Answer: D
Multiple-Choice Test Item
Which is the graph of
A
B
C
D
Answer: B
Example 1 Two Roots
Example 2 A Double Root
Example 3 No Real Roots
Example 4 Rational Roots
Example 5 Estimate Solutions to Solve a Problem
Solve
by graphing.
Graph the related function
The equation of the axis of symmetry is
or
or
When
f(x) equals
So the coordinates of the vertex are
Make a table of values to find other
points to sketch the graph.
x
–3
–1
0
1
2
3
4
6
f(x)
8
–6
–10
–12
–12
–10
–6
8
To solve
you need to know where
the value of f (x) is 0. This
occurs at the x-intercepts.
The x-intercepts of the
parabola appear to be
–2 and 5.
Check
Solve by factoring.
Original equation
Factor.
or
Zero Product Property
Solve for x.
Answer: The solutions of the equation are –2 and 5.
Solve
Answer: {–2, 4}
by graphing.
Solve
by graphing.
First rewrite the equation so one side is equal to zero.
Original equation
Add 9 to each side.
Simplify.
Graph the related function
x
1
2
f(x)
4
1
3
4
5
0
1
4
Notice that the vertex of the
parabola is the x-intercept.
Thus, one solution is 3.
What is the other solution?
Try solving the equation
by factoring.
Original equation
Factor.
or
Zero Product Property
Solve for x.
There are two identical factors for the quadratic function,
so there is only one root, called a double root.
Answer: The solution is 3.
Solve
Answer: {–1}
by graphing.
Solve
by graphing.
Graph the related function
x
f(x)
–3
–2
–1
6
3
2
0
1
3
6
Answer: The graph has no
x-intercept. Thus, there are no
real number solutions for the equation.
Solve
Answer: 
by graphing.
Solve
by graphing. If integral roots
cannot be found, estimate the roots by stating the
consecutive integers between which the roots lie.
Graph the related function
x
0
1
2
3
f(x)
2
–1
–2
–1
4
2
Notice that the value
of the function changes
from negative to positive
between the x values of
0 and 1 and between 3
and 4.
x
0
1
2
3
f(x)
2
–1
–2
–1
4
2
The x-intercepts of the graph are
between 0 and 1 and between 3 and 4.
Answer: One root is between 0 and 1,
and the other root is between 3 and 4.
Solve
Answer: One root is between
–1 and –2, and the other root is
between 3 and 4.
Model Rockets Shelly built a model rocket for
her science project. The equation
models the flight of the rocket, launched from ground
level at a velocity of 250 feet per second, where y is
the height of the rocket in feet after t seconds. For
how many seconds was Shelly’s rocket in the air?
You need to find the solution of the equation
Use a graphing calculator to graph
the related function
The x-intercept is
between 15 and 16 seconds.
Answer: between 15 and 16 seconds
Golf Martin hits a golf ball with an upward velocity
of 120 feet per second. The function
models the flight of the golf ball, hit at ground level,
where y is the height of the ball in feet after t
seconds. How long was the
golf ball in the air?
Answer: between 7 and
8 seconds
Example 1 Irrational Roots
Example 2 Complete the Square
Example 3 Solve an Equation by Completing the Square
Example 4 Solve a Quadratic Equation in Which a  1
Solve
by taking the square root of
each side. Round to the nearest tenth if necessary.
Original equation
is a perfect
square trinomial.
Take the square root of each side.
Simplify.
Definition of absolute value
Subtract 3 from each side.
Simplify.
Use a calculator to evaluate each value of x.
or
Answer: The solution set is {–5.2, –0.8}.
Solve
by taking the square root of
each side. Round to the nearest tenth if necessary.
Answer: {–2.3, –5.7}
Find the value of c that makes
a perfect square.
Method 1
Use algebra tiles.
is a perfect square.
Method 2
Complete the square.
Step 1
Find
Step 2
Square the result
of Step 1.
Step 3
Add the result of
Step 2 to
Answer:
Notice that
Find the value of c that makes
a perfect square.
Answer:
Solve
Step 1
by completing the square.
Isolate the x2 and x terms.
Original equation
Subtract 5 from
each side.
Simplify.
Step 2
Complete the square and solve.
Since
,
add 81 to each side.
Factor
Take the square root
of each side.
Add 9 to each side.
Simplify.
or
Check
Substitute each value for x in the
original equation.
Answer: The solution set is {1, 17}.
Solve
Answer: {–2, 10}
Boating Suppose the rate
of flow of an 80-foot-wide
river is given by the equation
where r is the rate in miles per hour, and x is the
distance from the shore in feet. Joacquim does not
want to paddle his canoe against a current faster than
5 miles per hour. At what distance from the river bank
must he paddle in order to avoid a current of 5 miles
per hour?
Explore
You know the function that relates distance
from shore to the rate of the river current. You
want to know how far away from the river
bank he must paddle to avoid the current.
Plan
Solve
Find the distance when
the square to solve
Use completing
Equation for
the current
Divide each side
by –0.01.
Simplify.
Since
add 1600 to each side.
Factor
Take the square root
of each side.
Add 40 to
each side.
Simplify.
Use a calculator to evaluate each value of x.
or
Examine
The solutions of the equation are about 7 ft
and about 73 ft. The solutions are distances
from one shore. Since the river is about 80 ft
wide,
Answer: He must stay within about 7 feet of either bank.
Boating Suppose the rate of flow of a 6-foot-wide
river is given by the equation
where r is the rate in miles per hour, and x is the
distance from the shore in feet. Joacquim does not
want to paddle his canoe against a current faster than
5 files per hour. At what distance from the river bank
must he paddle in order to avoid a current of 5 miles
per hour.
Answer: He must stay within 10 feet of either bank.
Example 1 Integral Roots
Example 2 Irrational Roots
Example 3 Use the Quadratic Formula to Solve
a Problem
Example 4 Use the Discriminant
Use two methods to solve
Method 1 Factoring
Original equation
Factor
or
Zero Product Property
Solve for x.
Method 2 Quadratic Formula
For this equation,
Quadratic Formula
Multiply.
Add.
Simplify.
or
Answer: The solution set is {–5, 7}.
Use two methods to solve
Answer: {–6, 5}
Solve
by using the Quadratic Formula.
Round to the nearest tenth if necessary.
Step 1
Rewrite the equation in standard form.
Original equation
Subtract 4 from
each side.
Simplify.
Step 2
Apply the Quadratic Formula.
Quadratic Formula
and
Multiply.
Add.
or
Check the solutions by using the CALC menu on a
graphing calculator to determine the zeros of the related
quadratic function.
Answer: The approximate solution
set is {–0.3, 0.8}.
Solve
by using the Quadratic Formula.
Round to the nearest tenth if necessary.
Answer: {–0.5, 0.7}
Space Travel Two possible future destinations of
astronauts are the planet Mars and a moon of the
planet Jupiter, Europa. The gravitational
acceleration on Mars is about 3.7 meters per
second squared. On Europa, it is only 1.3 meters
per second squared. Using the information and
equation from Example 3 on page 548 in your
textbook, find how much longer baseballs thrown
on Mars and on Europa will stay above the ground
than a similarly thrown baseball on Earth.
In order to find when the ball hits the ground, you must
find when H = 0. Write two equations to represent the
situation on Mars and on Europa.
Baseball Thrown on Mars
Baseball Thrown on Europa
These equations cannot be factored, and completing the
square would involve a lot of computation.
To find accurate solutions, use the Quadratic Formula.
Since a negative number of seconds is not reasonable,
use the positive solutions.
Answer: A ball thrown on Mars will stay aloft 5.6 – 2.2 or
about 3.4 seconds longer than the ball thrown on Earth.
The ball thrown on Europa will stay aloft 15.6 – 2.2 or
about 13.4 seconds longer than the ball thrown on Earth.
Space Travel The gravitational acceleration on
Venus is about 8.9 meters per second squared, and
on Callisto, one of Jupiter’s moons, it is 1.2 meters
per second squared. Suppose a baseball is thrown
on Callisto with an upward velocity of 10 meters per
second from two meters above the ground. Find
how much longer the ball will stay in air than a
similarly-thrown ball on Venus. Use the
equation
where H is the height
of an object t seconds after it is thrown upward, v is
the initial velocity, g is the gravitational pull, and h is
the initial height.
Answer: 16.9 – 2.4 or about 14.5 seconds
State the value of the discriminant for
Then determine the number of real roots of the
equation.
and
Simplify.
Answer: The discriminant is –220. Since the discriminant
is negative, the equation has no real roots.
.
State the value of the discriminant for
Then determine the number of real roots of the
equation.
Step 1
Rewrite the equation in standard form.
Original equation
Add 144 to
each side.
Simplify.
.
Step 2
Find the discriminant.
and
Simplify.
Answer: The discriminant is 0. Since the discriminant is 0,
the equation has one real root.
State the value of the discriminant for
Then determine the number of real roots of the
equation.
Step 1
.
Rewrite the equation in standard form.
Original equation
Subtract 12 from
each side.
Simplify.
Step 2
Find the discriminant.
and
Simplify.
Answer: The discriminant is 244. Since the discriminant
is positive, the equation has two real roots.
State the value of the discriminant for each
equation. Then determine the number of real
roots for the equation.
a.
Answer: –4; no real roots
b.
Answer: 0; 1 real root
c.
Answer: 120; 2 real root
Example 1 Graph an Exponential Function with a > 1
Example 2 Graph Exponential Functions with 0 < a < 1
Example 3 Use Exponential Functions to
Solve Problems
Example 4 Identify Exponential Behavior
State the y-intercept.
Graph
x
3x
–1
3–1
0
30
1
1
31
3
2
32
9
y
Graph the ordered pairs
and connect the points with
a smooth curve.
Answer: The y-intercept is 1.
Use the graph to determine the approximate
value of
The graph represents all real
values of x and their corresponding
values of y for
Answer: The value of y is about
5 when
Use a calculator to confirm
this value.
a. Graph
State the y-intercept.
Answer: The y-intercept is 1.
b. Use the graph
to determine the
approximate value
of
Answer: about 1.5
State the y-intercept.
Graph
x
y
–1
4
0
1
Graph the
ordered pairs
and connect the
points with a
smooth curve.
1
Answer: The y-intercept is 1.
Use the graph to determine the
approximate value of
Answer: The value of y is about
8 when
Use a calculator to confirm
this value.
a. Graph
State the y-intercept.
Answer: The y-intercept is 1.
b. Use the graph to
determine the approximate
value of
Answer: about 3
Depreciation People joke that the value of a new car
decreases as soon as it is driven off the dealer’s lot.
The function
models the
depreciation of the value of a new car that originally
cost $25,000. V represents the value of the car and t
represents the time in years from the time the car was
purchased. Graph the function. What values of V and t
are meaningful in the function.
Use a graphing calculator to graph the function.
Answer:
Only the values of V  25,000 and t  0 are meaningful
in the context of the problem.
What is the value of the car after one year?
Original equation
Use a calculator.
Answer: After one year, the car’s value is about $20,500.
What is the value of the car after five years?
Original equation
Use a calculator.
Answer: After five years, the car’s value is about $9270.
Depreciation The function
models
the depreciation of the value of a new car that
originally cost $22,000. V represents the value of the
car and t represents the time in years from the time
the car was purchased.
a. Graph the function.
What values of V
and t are meaningful
in the function.
Answer: Only the values
of V  22,000 and t  0
are meaningful in the
context of the problem.
b. What is the value of the car after one year?
Answer: $18,040
c. What is the value of the car after three years?
Answer: $12,130
Determine whether the set of data displays
exponential behavior.
x
y
0
10
10
25
20
62.5
30
156.25
Method 1 Look for a Pattern
The domain values are at regular intervals of 10.
Look for a common factor among the range values.
10
25
62.5
156.25
Answer: Since the domain values are at regular intervals
and the range values have a common factor, the data are
probably exponential. The equation for the data may
involve (2.5)x.
Method 2 Graph the Data
Answer: The graph shows a
rapidly increasing value of y
as x increases. This is a
characteristic of exponential
behavior.
Determine whether the set of data displays
exponential behavior.
x
y
0
10
10
25
20
40
30
55
Method 1 Look for a Pattern
The domain values are at regular intervals of 10.
The range values have a common difference of 15.
10
25
+15
40
+15
55
+15
Answer: Since the domain values are at regular intervals
and there is a common difference of 15, the data display
linear behavior.
Method 2 Graph the Data
Answer: The graph is a line,
not an exponential function.
Determine whether the set of data displays
exponential behavior.
a.
x
y
0
100
10
50
20
25
30
12.5
Answer: The domain values are at regular intervals and
the range values have a common factor of
so it is
probably exponential. Also the graph shows rapidly
decreasing values of y as x increases.
Determine whether the set of data displays
exponential behavior.
b.
x
y
0
–5
10
0
20
5
30
10
Answer: The domain values are at regular intervals and
the range values have a common difference of 5. The data
display linear behavior.
Example 1 Exponential Growth
Example 2 Compound Interest
Example 3 Exponential Decay
Example 4 Depreciation
Population In 2000, the United States had a
population of about 280 million, and a growth rate of
about 0.85% per year.
Write an equation to represent the population of the
United States since the year 2000.
General equation for
exponential growth
and
or 0.0085
Simplify.
Answer:
where y
represents the population and t represents the
number of years since 2000
According to the equation, what will be the
population of the United States in the year 2010?
Equation for the
population of the U.S.
or 10
Answer: In 2010, the population will be
about 304,731,295.
Population In 2000, Scioto School District had a
student population of about 4500 students, and a
growth rate of about 0.15% per year.
a. Write an equation to represent the student population
of the Scioto School District since the year 2000.
Answer:
b. According to the equation, what will be the student
population of the Scioto School District in the
year 2006?
Answer: about 4540 students
Compound Interest When Jing May was born, her
grandparents invested $1000 in a fixed rate savings
account at a rate of 7% compounded annually. The
money will go to Jing May when she turns 18 to help
with her college expenses. What amount of money will
Jing May receive from the investment?
Compound interest equation
Compound interest equation
Simplify.
Answer: She will receive about $3380.
Compound Interest When Lucy was 10 years old,
her father invested $2500 in a fixed rate savings
account at a rate of 8% compounded semiannually.
When Lucy turns 18, the money will help to buy her
a car. What amount of money will Lucy receive from
the investment?
Answer: about $4682
Charity During an economic recession, a charitable
organization found that its donations dropped by
1.1% per year. Before the recession, its donations
were $390,000.
Write an equation to represent the charity’s
donations since the beginning of the recession.
General equation for
exponential decay
Answer:
Simplify.
Charity During an economic recession, a charitable
organization found that its donations dropped by
1.1% per year. Before the recession, its donations
were $390,000.
Estimate the amount of the donations 5 years after
the start of the recession.
Equation for the amount
of donations
Answer: The amount of donations should be
about $369,017.
Charity A charitable organization found that the
value of its clothing donations dropped by 2.5% per
year. Before this downturn in donations, the
organization received clothing valued at $24,000.
a. Write an equation to represent the value of the
charity’s clothing donations since the beginning
of the downturn.
Answer:
b. Estimate the value of the clothing donations
3 years after the start of the downturn.
Answer: about $22,245
Depreciation Jackson and Elizabeth bought a house
when they first married ten years ago. Since that time
the value of the real estate in their neighborhood has
declined 3% per year. If they initially paid $179,000 for
their house, what is its value today?
General equation for
exponential decay
Simplify.
Use a calculator.
Answer: Their house will be worth about $131,999.
Depreciation A business owner
bought a computer system for the
office staff for $22,000. If the
computer system depreciates 8%
per year, find the value of the
computer system in 4 years.
Answer: about $15,761
Example 1 Recognize Geometric Sequences
Example 2 Continue Geometric Sequences
Example 3 Use Geometric Sequences to Solve a
Problem
Example 4 nth Term of a Geometric Sequence
Example 5 Find Geometric Means
Determine whether the sequence is geometric.
1, 4, 16, 64, 256, …
Determine the pattern.
1
4
16
64
256
In this sequence, each term in found by multiplying the
previous term by 4.
Answer: This sequence is geometric.
Determine whether the sequence is geometric.
1, 3, 5, 7, 9, 11, …
Determine the pattern.
1
3
5
7
9
11
In this sequence, each term is found by adding 2 to the
previous term.
Answer: This sequence is arithmetic, not geometric.
Determine whether each sequence is geometric.
a.
Answer: yes
b. 0, 11, 22, 33, 44, …
Answer: no
Find the next three terms in the geometric sequence.
20, –28, 39.2, …
Divide the second term by the first.
The common factor is –1.4. Use this information to find the
next three terms.
20, –28, 39.2
–54.88
76.832 –107.5648
Answer: The next three terms are –54.88,
76.832, and –107.5648.
Find the next three terms in the geometric sequence.
64, 48, 36, …
Divide the second term by the first.
The common factor is 0.8. Use this information to find the
next three terms.
64, 48, 36
27
20.25
15.1875
Answer: The next three terms are 27, 20.25,
and 15.1875.
Find the next three terms in each geometric sequence.
a. 42, 25.2, 15.12, …
Answer: 9.072, 5.4432, 3.26592
b. 100, –250, 625, …
Answer: –1562.5, 3906.25, –9765.625
Geography The population of the African country of
Liberia was about 2,900,000 in 1999. If the population
grows at a rate of about 5% per year, what will the
population be in the years 2003, 2004, and 2005?
The population is a geometric sequence in which the first
term is 2,900,000 and the common ratio is 1.05.
Year
Population
1999
2,900,000
2000
2,900,000(1.05) or 3,045,000
2001
3,045,000(1.05) or 3,197,250
2002
3,197,250(1.05) or 3,357,112.5
2003
3,357,112.5(1.05) or 3,524,968.1
2004
3,524,968.1(1.05) or 3,701,216.5
2005
3,701,216.5(1.05) or 3,886,277.3
Answer: The population of Liberia in the years 2003,
2004, and 2005 will be about 3,524,968, 3,701,217,
and 3,886,277, respectively.
Geography The population of the Baltic State of
Latvia was about 2,500,000 in 1998. If the population
grows at a rate of about 2% per year, what will the
population be in the years 2003 and 2004?
Answer: about 2,760,202 in 2003 and about 2,815,406
in 2004
Find the eighth term of a geometric sequence in
which
Formula for the nth term of
a geometric sequence
Answer: The eighth term in the sequence is 15,309.
Find the ninth term of a geometric sequence in
which
Answer: 786,432
Find the geometric mean in the sequence 7, ___, 112.
In the sequence,
you must first find r.
and
To find
Formula for the nth term of
a geometric sequence
Divide each side by 7.
Simplify.
Take the square root of each side.
If
the geometric mean is 7(4) or 28. If
the geometric mean is 7(–4) or –28.
Answer: The geometric mean is 28 or –28.
Find the geometric mean in the sequence 9, ___, 576.
Answer: 72 or –72
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