Summer 2012 LEAD Ambassador Team 1

134 Pages·2009·11.27 MB·English

Summer 2012
L.E.A.D. Ambassador Team 1
(Pre-Algebra & Algebra Course Content)

“Pre-Algebra”
Malloy, Molix-Bailey, Price, Willard
Glencoe McGraw-Hill (2008)
Functions and Graphing
7
(cid:129) Identify proportional or
nonproportional linear
relationships in problem situations
and solve problems.
(cid:129) Make connections among various
representations of a numerical
relationship.
(cid:129) Use graphs, tables, and algebraic
representations to make predictions
and solve problems.
Key Vocabulary
direct variation (p. 378)
function (p. 359)
rate of change (p. 371)
slope (p. 384)
Real-World Link
INSECTS If x is the number of chirps a cricket makes
every 15 seconds, the equation y = x + 40 can help
you estimate y, the outside temperature in degrees
Fahrenheit.
Functions and Graphing Make this Foldable to collect examples of functions and graphs. Begin with an
11’’ × 17’’ sheet of paper.
1 Fold the short sides so 2 Fold the top to the
they meet in the middle. bottom.
3 Open and Cut along 4 Add axes as shown. (cid:88)
the second fold to Label the quadrants (cid:8)(cid:12)(cid:11)(cid:9) (cid:8)(cid:11)(cid:12)(cid:11)(cid:9)
make four tabs. Staple on the tabs.
a sheet of grid paper (cid:47) (cid:87)
inside. (cid:8)(cid:12) (cid:9) (cid:8)(cid:11)(cid:12)(cid:115)(cid:9)
356 Chapter 7 Functions and Graphing
Cisca Casteljins/Foto Natura/Minden Pictures
7-1 Functions
Main Ideas
The table shows the time it should take a scuba
(cid:129) Determine whether Depth (ft) Time (s)
relations are diver to ascend to the surface from several depths
7.5 15
functions. to prevent decompression sickness.
15 30
(cid:129) Use functions to a. On grid paper, graph the depths and times as
22.5 45
describe relationships
ordered pairs (depth, time).
between two 30 60
quantities. b. Describe the relationship between the two sets
Source: diverssupport.com
of numbers.
New Vocabulary
c. I f a scuba diver is at 45 feet, what is the best estimate for the
function
amount of time she should take to ascend? Explain.
vertical line test
Relations and Functions Recall that a relation is a set of ordered pairs. A
function is a special relation in which each member of the domain is
paired with exactly one member in the range.
Function Not a Function
{(-2, 1), (-4, 3), (-5, 4), (-9, 7)} {(-2, 1), (-2, 3), (-5, 4), (-9, 7)}
(cid:19)(cid:211) (cid:163) (cid:19)(cid:211) (cid:163)
(cid:19)(cid:123) (cid:206) (cid:206)
(cid:19)(cid:120) (cid:123) (cid:19)(cid:120) (cid:123)
(cid:19)(cid:153) (cid:199) (cid:19)(cid:153) (cid:199)
Review This is a function because each domain value is This is not a function because -2 in the domain
Vocabulary paired with exactly one range value is paired with two range values, 1 and 3.
domain the set of Since functions are relations, they can be represented using ordered pairs,
x-coordinates in a tables, or graphs.
relation; Example: The
domain of {(1, 4),
EXAMPLE Ordered Pairs and Tables as Functions
(-3, -7)} is {1, -3}.
range the set of
Determine whether each relation is a function. Explain.
y-coordinates in a
relation; Example: The a. {(-3, 1), (-2, 4), (-1, 7), (0, 10), (1, 13)}
range of {(1, 4),
(-3, -7)} is This relation is a function because each element of the domain is
{4, -7}. (Lesson 1-6) paired with exactly one element of the range.
b. x 5 3 2 0 -4 -6 This is a function because for each
y 1 3 1 3 -2 2 element of the domain, there is
only one corresponding element
in the range.
1A. {(5, 1), (6, 3), (7, 5), (8, 0)} 1B.
x -1 -6 -3 -1 -5 -2
y 7 6 2 8 -2 1
Personal Tutor at pre-alg.com
Lesson 7-1 Functions 359
Another way to determine whether a relation
y
is a function is to apply the vertical line test
to the graph of the relation. Use a pencil or
Vocabulary Link straightedge to represent a vertical line.
Function
Everyday Use a Place the pencil at the left of the graph. Move O x
relationship in which one it to the right across the graph. If, for each
quality or trait depends on
value of x in the domain, it passes through no
another. Height is a
more than one point on the graph, then the
function of age.
graph represents a function.
Math Use a relationship
in which a range value
depends on a domain
EXAMPLE Use a Graph to Identify Functions
value, y is a function of x.
Determine whether the graph at the right is a y
function. Explain your answer.
The graph represents a relation that is not a
function because it does not pass the vertical line
O x
test. At least one input value has more than one
output value. By examining the graph, you can see
that when x = 2, there are three different y values.
2. Determine whether the graph of times and distances below is a function.
Explain your answer.
Describe Relationships A function describes the relationship between two
quantities such as time and distance. For example, the distance you travel on a
bike depends on how long you ride the bike. In other words, distance is a
function of time.
Time Distance 9 y
(min) (m) 8
4 1 mi) 7
6
12 3 e ( 5
c
n 4
16 4 a
st 3
20 5 Di 2
1
28 7 x
0 4 812162024283236
Time (min)
SCUBA DIVING The table shows the water pressure
Water Pressure
Depth (ft)
as a scuba diver descends. (lb/ft2)
a. Do these data represent a function? Explain. 0 0
1 62.4
Real-World Link This relation is a function because at each depth,
2 124.8
there is only one measure of pressure.
Most of the ocean’s
3 187.2
marine life and coral
b. Describe how water pressure is related to depth. 4 249.6
live and grow within
30 feet of the surface. Water pressure depends on the depth. As the 5 312.0
Source: scuba.about.com depth increases, the pressure increases. Source: infoplease.com
360 Chapter 7 Functions and Graphing Extra Examples at pre-alg.com
Peter/Stef Lamberti/Getty Images
3. SALES Do these data in the table represent a function? Describe how price
is related to the number of balloons purchased.
Number of Balloons 100 200 300 400 500
Price per Balloon $0.99 $0.90 $0.79 $0.60 $0.50
Determine whether each relation is a function. Explain.
Example 1 1. {(13, 5), (-4, 12), (6, 0), (13, 10)} 2. {(9.2, 7), (9.4, 11), (9.5, 9.5), (9.8, 8)}
(p. 359)
3. 4.
Domain Range x y
-3 3 5 4
-1 -2 2 8
0 5 -7 9
1 -4 2 12
2 3 5 14
Example 2 5. y 6. y
(p. 360)
O x O x
Example 3 MEASUREMENTS For Exercises 7 and 8, use the
Foot Length Height
Name
(pp. 360–361) data in the table. (cm) (cm)
7. Do the data represent a function? Explain. Remana 24 163
8. Is there any relation between foot length and Enrico 25 163
height? Jahad 24 168
Cory 26 172
HOMEWORK HELP Determine whether each relation is a function. Explain.
For See 9. {(-1, 6), (4, 2), (2, 36), (1, 6)} 10. {(-2, 3), (4, 7), (24, -6), (5, 4)}
Exercises Examples
11. {(9, 18), (0, 36), (6, 21), (6, 22)} 12. {(5, -4), (-2, 3), (5, -1), (2, 3)}
9–16 1
17–20 2 13. 14.
Domain Range Domain Range
21–24 3 -4 -2 -1 5
-2 1 -2 5
0 2 -2 1
3 1 -6 1
15. 16.
x y x y
-7 2 14 5
0 4 15 10
11 6 16 15
11 8 17 20
0 10 18 25
Lesson 7-1 Functions 361
Determine whether each relation is a function. Explain.
17. y 18. y
O x O x
19. y 20. y
O x O x
FARMING For Exercises 21–24, use Farms in the United States
the table that shows the number
Number Average Size
Year
and size of farms in the United (millions) (acres)
States every decade from 1950 1950 5.6 213
to 2000. 1960 4.0 297
21. Is the relation (year, number of 1970 2.9 374
farms) a function? Explain. 1980 2.4 426
22. Describe how the number of 1990 2.1 460
farms is related to the year. 2000 2.2 434
23. Is the relation (number of
Real-World Link farms, average size of farms) a
The lowest wind chill function? Explain.
Source: U.S. Dept. of Agriculture
temperature ever
24. Describe how the average size
recorded at an NFL
game was -59°F in of farms is related to the year.
Cincinnati, Ohio, on
January 10, 1982. Tell whether each statement is always, sometimes, or never true. Explain.
Source: Southern AER 25. A function is a relation. 26. A relation is a function.
ANALYZE TABLES For Exercises 27 and 28, use the
Wind Speed Wind Chill
table that shows how various wind speeds affect (mph) Temperature (°F)
the actual temperature of 15°F. 0 15
27. Do the data represent a function? Explain. 10 3
28. Describe how wind chill temperatures are 20 -2
related to wind speed. 30 -5
EXTRA PRACTTICEE
40 -8
See pages 776, 800. 29. RESEARCH Use the Internet or another source to
Source: National Weather Service
find the complete wind chill table. Does the data
Self-Check Quiz at for actual temperature and wind chill temperature
pre-alg.com
for a specific wind speed represent a function? Explain.
H.O.T. Problems 30. OPEN ENDED Draw the graph of a relation that is not a function. Explain
why it is not a function.
31. REASONING Describe three ways to represent a function. Show an example
of each. Then describe three ways to represent a relation that is not a
function and show an example of each.
362 Chapter 7 Functions and Graphing
Craig Tuttle/CORBIS
CHALLENGE The inverse of any relation is obtained by switching the
coordinates in each ordered pair of the relation.
32. Determine whether the inverse of the relation {(4, 0), (5, 1), (6, 2), (6, 3)} is
a function.
33. Is the inverse of a function always, sometimes, or never a function? Give an
example to explain your reasoning.
34. Writing in Math How can the relationship between water depth and
time to ascend to the water’s surface be a function? Include a discussion
about whether water depth can ever have two corresponding times to
ascend to the water’s surface.
35. Which statement is true about the 36. The table shows the water
data in the table? temperatures at various depths in a
lake. Describe how temperature is
A The data represent a
x y
related to the depth.
function.
-4 -4
B The data do not 2 16 Depth (ft) 0 10 20 30 40 50
represent a function. 5 8 Temperature (°F ) 74 72 71 61 55 53
10 -4
C As the value of x F The water temperature stays the
12 15
increases, the value of y same as the depth increases.
increases.
G The water temperature decreases as
D A graph of the data would not pass the depth increases.
the vertical line test.
H The water temperature increases as
the depth increases.
J The water temperature decreases as
the depth decreases.
37. TECHNOLOGY The table shows the results of a survey in which Activity Percent
middle school students were asked whether they ever used E-mail 71%
the Internet for the activities listed. Use the data to predict research for school 70%
how many students in a middle school of 650 have used the instant message 61%
Internet to do research for school. (Lesson 6-10) games 71%
Source: Atlantic Research and Consulting
Find each percent of change. Round to the nearest tenth, if necessary. Then
state whether the percent of change is a percent of increase or a percent of
decrease. (Lesson 6-9)
38. from $56 to $49 39. from 110 mg to 165 mg
40. SCIENCE The length of a DNA strand is 0.0000007 meter. Write the length
of a DNA strand using scientific notation. (Lesson 4-7)
Evaluate each expression if x = 4 and y = -1. (Lesson 1-3)
41. 3x + 1 42. 2y 43. y + 6 44. 20 - 4x
Lesson 7-1 Functions 363
7-2 Representing Linear
Functions
Main Ideas
Interactive Lab pre-alg.com
(cid:129) Solve linear equations
Peaches cost $1.50 per can.
with two variables. Number of 1.50x Cost (y)
a. C omplete the table to Cans (x)
(cid:129) Graph linear
equations using find the cost of 2, 3, 1 1.50(1) 1.50
ordered pairs. and 4 cans of peaches. 2
New Vocabulary b. O n grid paper, graph 3
the ordered pairs 4
linear equation (number, cost). Then
draw a line through the points.
c. W rite an equation representing the relationship between
number of cans x and cost y.
Solutions of Equations An equation such as y = 1.50x is called a linear
equation. A linear equation in two variables is an equation in which the
variables appear in separate terms and neither variable contains an
exponent other than 1.
Solutions of a linear equation are ordered pairs that make the equation
Reading Math true. One way to find solutions is to make a table. Consider y = -x + 8.
Input and Output
The variable for the y = -x + 8
input is called the
x y = -x + 8 y (x, y)
independent variable
because the values are ⎧ -1 y = -(-1) + 8 9 (-1, 9) ⎫
chosen and do not Step 1 Choose (cid:3) (cid:3) 0 y = -(0) + 8 8 (0, 8) (cid:3) (cid:3) Step 4 Write the
depend upon the other any convenient ⎨ ⎬ solutions as
variable. The variable (cid:3) 1 y = -(1) + 8 7 (1, 7) (cid:3)
values for x. ordered pairs.
ftohre tdheep oeuntdpeunt ti sv caarlilaebdl e (cid:3) ⎩ 2 y = -(2) + 8 6 (2, 6) (cid:3) ⎭
because it depends on
the input value.
Step 2 Substitute Step 3 Simplify to
the values for x. find the y-values.
So, four solutions of y = -x + 8 are (-1, 9), (0, 8), (1, 7), and (2, 6).
EXAMPLE Use a Table of Ordered Pairs
Find four solutions of y = 2x - 1.
Choose four values for x. Then x y = 2x - 1 y (x, y)
substitute each value into the equation 0 y = 2(0) - 1 (cid:2)1 (0, -1)
and solve for y. 1 y = 2(1) - 1 1 (1, 1)
Four solutions are (0, -1), (1, 1), (2, 3), 2 y = 2(2) - 1 3 (2, 3)
and (3, 5). 3 y = 2(3) - 1 5 (3, 5)
1. Find four solutions of y = x + 5.
Lesson 7-2 Representing Linear Functions 365
Solve an Equation for y
CELL PHONES Games cost $8 to download onto a cell phone. Ring tones
cost $1. Find four solutions of 8x + y = 20 in terms of the numbers of
games x and ring tones y Darcy can buy with $20. Explain each solution.
First, rewrite the equation by solving for y.
8x + y = 20 Write the equation.
8x + y - 8x = 20 - 8x Subtract 8x from each side.
y = 20 - 8x Simplify.
Choose four x values and substitute them
x y = 20 - 8x y (x, y)
into y = 20 - 8x.
1 y = 20 - 8(1) 12 (1, 12)
Choosing
x-Values (1, 12) → She can buy 1 game and 2 y = 20 - 8(2) 4 (2, 4)
It is often convenient 12 ring tones. _ 1 y = 20 - 8( _ 1 ) 18 ( _ 1 , 18)
to choose 0 as an x (2, 4) → She can buy 2 games and 4 4 4
value to find a value 5 y = 20 - 8(5) -20 (5, -20)
4 ring tones.
for y.
(_1 )
, 18 → T his solution does not make sense in the situation because there
4
cannot be a fractional number of games.
(5, -20) → T his solution does not make sense in the situation because there
cannot be a negative number of ring tones.
2. SHOPPING Fancy goldfish x cost $3, and regular goldfish y cost $1. Find
three solutions of 3x + y = 8 in terms of the number of each type of fish
Tyler can buy for $8. Describe what each solution means.
Personal Tutor at pre-alg.com
Graph Linear Equations A linear equation can also be represented by a graph.
Linear Equations
Reading Math
y y y
Linear Equations
Graphs of all “linear” y (cid:2) 1 x
equations are straight 3
lines. The coordinates
of all points on a line
O x O x O x
are solutions of the
equation.
y (cid:2) x (cid:3) 1
y (cid:2) (cid:4)2x

Nonlinear Equations
y y y
y (cid:2) x2 (cid:3) 1 y (cid:2) (cid:4)2x3
O x O x O x
y (cid:2) 3
x

366 Chapter 7 Functions and Graphing
EXAMPLE Graph a Linear Equation
Graph y = x + 1 by plotting ordered pairs.
x y = x + 1 y (x, y)
First, find ordered pair solutions. Four -1 y = -1 + 1 0 (-1, 0)
solutions are (-1, 0), (0, 1), (1, 2), and (2, 3). 0 y = 0 + 1 1 (0, 1)
1 y = 1 + 1 2 (1, 2)
2 y = 2 + 1 3 (2, 3)
Plot these ordered pairs and draw a line y
through them. Note that the ordered pair for
Plotting Points any point on this line is a solution of y = x + 1. (2, 3)
(1, 2)
It is best to find at least The line is a complete graph of the function. ((cid:4)1, 0) (0, 1)
three points. You can CHECK It appears from the graph that O x
also graph just two
(-2, -1) is also a solution. Check
points to draw the line y (cid:2) x (cid:3) 1
and then graph one this by substitution.
point to check.
y = x + 1 Write the equation.
-1 (cid:10) -2 + 1 Replace x with -2 and y with -1.
-1 = -1 (cid:2) Simplify.
3. Graph y = 2x - 1 by plotting ordered pairs.
A linear equation is one of many ways to represent a function.
Representing Functions
Words The value of y is 3 less than the corresponding value of x.
Table of Graph y
x y
Ordered Pairs
0 -3
1 -2 O x
2 -1
3 0 y (cid:2) x (cid:4) 3
Equation y = x - 3
Example 1 Find four solutions of each equation. Show each solution in a table of
(p. 365) ordered pairs.
1. y = x + 8 2. y = 4x 3. y = 2x - 7 4. -5x + y = 6
Example 2 5. SCIENCE The distance in miles d that light travels in t seconds is given by
(p. 366) the linear function d = 186,000t. Find two solutions of this equation and
describe what they mean.
Example 3 Graph each equation by plotting ordered pairs.
(p. 367)
6. y = x + 3 7. y = 2x - 1 8. x + y = 5
Extra Examples at pre-alg.com Lesson 7-2 Representing Linear Functions 367