Transcript Answer - Dougher-Algebra2-2009-2010

Algebra 2 Interactive Chalkboard
Copyright © by The McGraw-Hill Companies, Inc.
Send all inquiries to:
GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 2-1 Relations and Functions
Lesson 2-2 Linear Equations
Lesson 2-3 Slope
Lesson 2-4 Writing Linear Equations
Lesson 2-5 Modeling Real-World Data: Using
Scatter Plots
Lesson 2-6 Special Functions
Lesson 2-7 Graphing Inequalities
Example 1 Domain and Range
Example 2 Vertical Line Test
Example 3 Graph Is a Line
Example 4 Graph Is a Curve
Example 5 Evaluate a Function
Click the mouse button or press the
Space Bar to display the answers.
Relation
A relation is a set of
inputs and outputs,
often written as ordered
pairs (input, output).
We can also represent a
relation as a mapping
diagram or a graph. For
example, the relation
can be represented as:
Mapping Diagram of Relation
Domain & Range
The first components
in the ordered pairs (xcoordinate) make up
the domain.
The second
components (ycoordinate or f(x)
coordinate) make up
the range.
Domain & Range
For example, the group
of ordered pairs (1, 2),
(3, 5), and
(7, 10) is a relation.
The Domain of this
relation consists of the
x-coordinates of the
ordered pairs and the
range consists of the
y-coordinates.
Domain: (1, 3, 7}
Range: {2, 5, 10}
Function
A function is a relation in which each input
has only one output.
For example, the relation {(1, 2), (3, 5), and
(7, 10)} is a function since no x values
produce more than one y value.
The relation {(1, 2), (3, 4), and (1, 5)} would
not be considered a function since the x value
1 produces two different y values, 2 and 5.
Functions
• The diagram on the
right shows the
relation {(-1, 2), (0, 2),
(1, 3), (2, -2), (8, 3)}.
• This relation is a
function since each x
value produces one
and only one y value.
The Vertical Line Test
• To determine if a relation
is a function you can use
the vertical line test. A
relation is a function if
there are no vertical lines
that intersect the graph at
more than one point.
• The graph at the right is
not a function since it
would fail the vertical line
test.
Function
Not a Function
State the domain and range
of the relation shown in the
graph. Is the relation a
function?
The relation is {(1, 2), (3, 3),
(0, –2), (–4, 0), (–3, 1)}.
Answer: The domain is
{–4, –3, 0, 1, 3}. The range
is {–2, 0, 1, 2, 3}. Each member
of the domain is paired with exactly one member of the
range, so this relation is a function.
State the domain and range
of the relation shown in the
graph. Is the relation a
function?
Answer: The domain is
{–3, 0, 2, 3}. The range
is {–2, –1, 0, 1}. Yes, the
relation is a function.
Transportation The table
shows the average fuel
efficiency in miles per
gallon for light trucks for
several years. Graph this
information and
determine whether it
represents a function.
Year
Fuel Efficiency
(mi/gal)
1995
20.5
1996
1997
1998
1999
20.8
20.6
20.9
20.5
2000
2001
20.5
20.4
Year
Fuel Efficiency
(mi/gal)
1995
20.5
1996
1997
1998
1999
20.8
20.6
20.9
20.5
2000
2001
20.5
20.4
Use the vertical line test. Notice that no vertical
line can be drawn that contains more than one
of the data points.
Answer: Yes, this relation is a function.
Health The table shows
the average weight of a
baby for several months
during the first year.
Graph this information
and determine whether it
represents a function.
Age
(months)
Weight
(pounds)
1
12.5
2
4
6
9
16
22
24
25
12
26
Answer:
Yes, this relation is a function.
Graph the relation represented by
Make a table of values to find ordered pairs that satisfy the
equation. Choose values for x and find the corresponding
values for y. Then graph the ordered pairs.
x
y
–1
0
1
2
–4
–1
2
5
(2, 5)
(1, 2)
(0, –1)
(–1, –4)
Find the domain and range.
Since x can be any real number,
there is an infinite number of
ordered pairs that can be
graphed. All of them lie on the
line shown. Notice that every
real number is the x-coordinate
of some point on the line. Also,
every real number is the
y-coordinate of some point
on the line.
(2, 5)
(1, 2)
(0, –1)
(–1, –4)
Answer: The domain and range are both all
real numbers.
Determine whether the relation is a function.
This graph passes the vertical
line test. For each x value, there
is exactly one y value.
(2, 5)
(1, 2)
Answer: Yes, the equation
represents
a function.
(0, –1)
(–1, –4)
a. Graph
Answer:
b. Find the domain and range.
Answer: The domain and range
are both all real numbers.
c. Determine whether the
relation is a function.
Answer: Yes, the equation
is a function.
Graph the relation represented by
Make a table. In this case, it is easier to choose y values
and then find the corresponding values for x. Then sketch
the graph, connecting the points with a smooth curve.
x
y
5
2
1
2
5
–2
–1
0
1
2
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
Find the domain and range.
Every real number is the
y-coordinate of some point on
the graph, so the range is all real
numbers. But, only real numbers
that are greater than or equal to
1 are x-coordinates of points on
the graph.
Answer: The domain is
The range is all
real numbers.
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
.
Determine whether the relation is a function.
x
y
5
2
1
2
5
–2
–1
0
1
2
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
You can see from the table and the vertical line test that
there are two y values for each x value except x = 1.
Answer: The equation
a function.
does not represent
a. Graph
Answer:
b. Find the domain and range.
Answer: The domain is
{x|x  –3}. The range
is all real numbers.
c. Determine whether the
relation is a function.
Answer: No, the equation
represent a function.
does not
Given
, find
Original function
Substitute.
Simplify.
Answer:
Given
find
Original
function
Substitute.
Multiply.
Simplify.
Answer:
Given
, find
Original function
Substitute.
Answer:
Given
find each value.
and
a.
Answer: 6
b.
Answer: 0.625
c.
Answer:
Class Work:
Page 60-61 in your textbooks
#18, 20, 22, 28, 30, 35-41
Assignment:
Page 60-61 #24, 32, 46, 47
Example 1 Identify Linear Functions
Example 2 Evaluate a Linear Function
Example 3 Standard Form
Example 4 Use Intercepts to Graph a Line
Linear Equations
• Looks like:
•
•
•
•
x + 5 = 10
x+y=1
3x – 5y = 12
A linear equation has no operations other than
addition, subtraction, and multiplication of a variable
by a constant.
The variables may not be multiplied together or appear
in a denominator.
A linear equation does not contain variables with
exponents other than 1.
The graph of a linear equation is always a straight line.
Linear Functions
• A function whose ordered pairs satisfy a
linear equation.
• Linear functions look like:
f(x) = x + 3
f(x) = 5x + 1
f(x) = 2x – 4
Linear Functions
• Look at the linear function f(x) = x – 7.
• This means that you may plug in some
value for x, and subtract 7 from it to get
f(x). Note: f(x) acts as y.
State whether
Explain.
is a linear function.
Answer: This is a linear function because it is in the form
State whether
Explain.
is a linear function.
Answer: This is not a linear function because x has an
exponent other than 1.
State whether
Explain.
is a linear function.
Answer: This is a linear function because it can be written
as
State whether each function is a linear function.
Explain.
a.
Answer: yes;
b.
Answer: No; x has an exponent
other than 1.
c.
Answer: No; two variables are
multiplied together.
Try These
State whether each function is linear. Write yes or no. If no, explain your
reasoning.
h(x) = 2x3 –4x2 + 5
f(x) = 6x - 19
Try These
State whether each function is linear. Write yes or no. If no, explain your
reasoning.
h(x) = 2x3 –4x2 + 5
No, a linear function
No, a linear function
cannot have a variable cannot have variables
in the denominator.
raised to powers greater
than 1.
f(x) = 6x - 19
Yes
No, a linear function
cannot have square
roots.
Meteorology The linear function
can be used to find the number of degrees Fahrenheit,
f (C), that are equivalent to a given number of degrees
Celsius, C.
On the Celsius scale, normal body temperature is
37C. What is normal body temperature in degrees
Fahrenheit?
Original function
Substitute.
Simplify.
Answer: Normal body temperature, in degrees
Fahrenheit, is 98.6F.
There are 100 Celsius degrees between the freezing
and boiling points of water and 180 Fahrenheit
degrees between these two points. How many
Fahrenheit degrees equal 1 Celsius degree?
Divide 180 Fahrenheit degrees by 100 Celsius degrees.
Answer: 1.8F = 1C
Meteorology The linear function
can be
used to find the distance d(s) in miles from a storm,
based on the number of seconds s that it takes to
hear thunder after seeing lightning.
a. If you hear thunder 10 seconds after seeing lightning,
how far away is the storm?
Answer: 2 miles
b. If the storm is 3 miles away, how long will it take to
hear thunder after seeing lightning?
Answer: 15 seconds
Try These
When a sound travels through water, the distance
y in meters that the sounds travels in x seconds
is given by the equation y = 1440x.
a) How far does sounds travel underwater in 5
seconds?
b) In air, the equation is y = 343x. Does sound travel
faster in air or water? Explain.
Try These
When a sound travels through water, the distance
y in meters that the sounds travels in x seconds
is given by the equation y = 1440x.
a) How far does sounds travel underwater in 5
seconds?
7200 meters
b) In air, the equation is y = 343x. Does sound travel
faster in air or water? Explain.
Sound travels only 1715 meters in 5 seconds in
air, so it travels faster under water.
Standard Form of a Linear
Equation
Ax + By = C
Where A, B, and C are integers whose
greatest common factor is 1.
Example:
5x + 3y = 10
Write
in standard form. Identify A, B, and C.
Original equation
Subtract 3x from each side.
Multiply each side by –1 so
that A  0.
Answer:
and
Write
and C.
in standard form. Identify A, B,
Original equation
Subtract 2y from each side.
Multiply each side by –3 so that
the coefficients are all integers.
Answer:
and
Write
and C.
in standard form. Identify A, B,
Original equation
Subtract 4 from each side.
Divide each side by 2 so that the
coefficients have a GCF of 1.
Answer:
and
Write each equation in standard form. Identify A, B,
and C.
a.
Answer:
and
b.
Answer:
and
c.
Answer:
and
Try These
Write each equation in standard form. Identify A, B, and C.
y = 12x
4x = 8y – 12
x = 7y + 2
Try These
Write each equation in standard form. Identify A, B, and C.
y = 12x
x = 7y + 2
12x – y = 0;
x – 7y = 2;
A = 12, B = -1, C = 0
A = 1, B = -7, C = 2
4x = 8y – 12
x – 2y = -3;
x – y = -6;
A = 1, B = -2, C = -3
A = 1, B = -1, C = -6
X-Intercept
•
The x-intercept is the
place at which a graph
crosses the x-axis.
At the right is the graph
of the linear function of y
• graph
= 2x + 4. In this
the x –intercept is –2. In
order pair form it is
(-2, 0)
Y-Intercept
•
The y-intercept is the
place at which a
graph crosses the yaxis.
In this graph• the y –
intercept is 4. In
order pair form it is
(0, 4)
Finding x- and y-intercepts
• Since the x-intercept is the
place where the graph
crosses the x-axis, it is
also the place where the yvalue is 0. So, to find the
x-intercept of an equation,
just plug in 0 for y. And
to find the y-intercept,
plug in 0 for x.
Example: Find the x- and
y-intercepts of
2x + 3y = 12.
x-intercept:
2x + 3(0) = 12
2x = 12
X=6
Y-intercept:
2(0) + 3y = 12
3y = 12
Y=4
Finding x- and y-intercepts
Continued
• So this means that our graph hits the x-axis
at 6, also known as (6, 0) and hits the y-axis
at 4, also known as (0, 4).
• The graph of this equation could easily be
drawn now that we know two points:
4
6
Find the x-intercept and the y-intercept of the graph of
Then graph the equation.
The x-intercept is the value of x when
Original equation
Substitute 0 for y.
Add 4 to each side.
Divide each side by –2.
The x-intercept is –2. The graph crosses the
x-axis at (–2, 0).
Likewise, the y-intercept is the value of y when
Original equation
Substitute 0 for x.
Add 4 to each side.
The y-intercept is 4. The graph crosses the y-axis at (0, 4).
Use the ordered pairs to graph this equation.
Answer: The x-intercept is –2, and the y-intercept is 4.
(0, 4)
(–2, 0)
Find the x-intercept and the y-intercept of the graph of
Then graph the equation.
Answer: The x-intercept is –2, and the y-intercept is 6.
Try These
Find the x-intercept and y-intercept of the graph of each equation. Then
graph the equation.
2x – 6y = 12
y = 4x – 2
Try These
Find the x-intercept and y-intercept of the graph of each equation. Then
graph the equation.
2x – 6y = 12
x-intercept: 6
y-intercept: -2
y = 4x – 2
x-intercept: 1/2
y-intercept: -2
Assignment:
Page 66-67 #42, 53, 54, 55
Example 1 Find Slope
Example 2 Use Slope to Graph a Line
Example 3 Rate of Change
Example 4 Parallel Lines
Example 5 Perpendicular Line
Slope of a Line
• The slope of a line measures the steepness of the
line. Most of you are probably familiar with
associating slope with "rise over run".
• Rise means how many units you move up or down
from point to point. On the graph that would be a
change in the y values. Run means how far left
or right you move from point to point. On the
graph, that would mean a change of x values.
Positive Slope
• Here are some
visuals to help you
with this definition:
The slope of this line is 2/3.
Negative Slope
• Here is a graph of a
line with a negative
slope. The slope here
is –2/1.
Formula for Calculating
Slope of a Line
• To calculate the slope of a line given two
points on that line you can use the formula:
Find the slope of the line that passes through (1, 3)
and (–2, –3). Then graph the line.
Slope formula
and
Simplify.
Graph the two ordered pairs and draw the line.
Use the slope to check your
graph by selecting any point
on the line. Then go up 2 units
and right 1 unit or go down 2
units and left 1 unit. This point
should also be on the line.
Answer: The slope of the
line is 2.
(1, 3)
(–2, –3)
Find the slope of the line that passes through (2, 3)
and (–1, 5). Then graph the line.
Answer: The slope of the
line is
Try These
Find the slope of the line that passes through each pair of points.
(6, 8), (5, -5)
(-2, -3), (0, -5)
(4, -1.5), (4, 4.5)
1 2 51
( , ), (
)
2 3 6, 4
Try These
Find the slope of the line that passes through each pair of points.
(6, 8), (5, -5)
Slope = 13
(-2, -3), (0, -5)
Slope = -1
(4, -1.5), (4, 4.5)
1 2 51
( , ), (
)
2 3 6, 4
Slope is undefined
Slope = -5/4
Graph the line passing through (1, –3) with a slope
of
Graph the ordered pair (1, –3).
Then, according to the slope,
go down 3 units and right 4
units. Plot the new point at
(5, –6).
Draw the line containing
the points.
(1, –3)
(5, –6)
Graph the line passing through (2, 5) with a slope
of –3.
Answer:
Try These
Graph the line passing through the given point with the given slope.
(-3, -1), m = -1/5
(3, -4), m = 2
(2, 2), m = 0
Try These
Graph the line passing through the given point with the given slope.
(-3, -1), m = -1/5
(3, -4), m = 2
(2, 2), m = 0
Communication Refer to
the graph. Find the rate of
change of the number of
radio stations on the air in
the United States from 1990
to 1998.
Slope
formula
Substitute.
Simplify.
Answer: Between 1990 and 1998, the number of radio
stations on the air in the United States increased at an
average rate of 0.225(1000) or 225 stations per year.
Computers Refer to the
graph. Find the rate of
change of the number of
households with computers
in the United States from
1984 to 1998.
Answer: The rate of change
is 2.9 million households
per year.
Slopes of Parallel Lines
• Parallel lines have the
same slopes. This is
because the are going
in the same direction.
They do, however,
have different yintercepts since they
do not hit the y-axis at
the same place.
Slopes of Perpendicular Lines
• Perpendicular lines
have opposite
reciprocal slopes. For
example, 2/3 and –3/2.
• The graph at the right
shows a pair of
perpendicular liens
whose slopes are 1/1
and –1/1.
Graph the line through (1, –2) that is parallel to the
line with the equation
The x-intercept is –2 and the
y-intercept is 2.
Use the intercepts to graph
The line rises 1 unit for every
1 unit it moves to the right, so
the slope is 1.
Now, use the slope and the
point at (1, –2) to graph the
line parallel to
(2, –1)
(1, –2)
Graph the line through (2, 3) that is parallel to the
line with the equation
Answer:
Graph the line through (2, 1) that is perpendicular
to the line with the equation
The x-intercept is
or 1.5
and the y-intercept is –1.
Use the intercepts to graph
2x – 3y = 3
The line rises 1 unit for every
1.5 units it moves to the right,
so the slope is
or
Graph the line through (2, 1) that is perpendicular
to the line with the equation
The slope of the line
perpendicular is the opposite
reciprocal of
or
Start at (2, 1) and go down
3 units and right 2 units.
Use this point and (2, 1) to
graph the line.
(2, 1)
2x – 3y = 3
(4, –2)
Graph the line through (–3, 1) that is perpendicular
to the line with the equation
Answer:
Try These
Graph the line that satisfies each set of conditions.
Passes through (2, -2) and
is parallel to a line whose
slope is –1.
Passes through (3, 0) and
is perpendicular to a line
whose slope is ¾.
Passes through the origin
and is parallel to the graph
of x + y = 10
Try These
Graph the line that satisfies each set of conditions.
Passes through (2, -2) and
is parallel to a line whose
slope is –1.
Passes through (3, 0) and
is perpendicular to a line
whose slope is ¾.
Passes through the origin
and is parallel to the graph
of x + y = 10
Assignment:
Page 72-73 #18, 34, 44, 46