We focus on four ways to mathematically describe linear relationships between
two quantities .
| 1. Equations in Two Variables |
| The general form of the equation of a line is Ax + By =
C. Two very useful forms of the equation of a line are the
slope- intercept form and the point- slope form . |
1. Write the equation of a line with a slope of - 3 and
a y-intercept of (0, -4). Answer: y = -3x - 4
2. Write the equation of
the line that passes through (5, 2) and (-5, 0). Answer in
slope-intercept form. Answer: |
| 2. Rectangular Coordinate Graphs |
| The graph of an equation is a "picture" of all of its
solutions (x, y). Important information can be obtained from a graph. |
3. Complete the table of solutions for 2x - 4y = 8. Then
graph the equation.
4. See the graph below
a. What information does the y-intercept of the graph give us?
Answer: When new, the press cost $40,000
b. What is the slope of the line and what does it tell us? Answer:
-5,000; the value of the press decreased $5,000/yr.
|
| 3. Linear Functions |
| We can use the notation f(x) = mx + b to describe linear
functions. |
5. The function f(x) = 25 + 35x gives the cost (in
dollars) to rent a cement mixer for x days. Find f (3). What does it
represent? Answer: 130; the cost to rent the mixer for 3 days
6. The
function T(c) = 1/4c+40 predicts the outdoor temperature T in degrees
Fahrenheit using the number of cricket chirps c per minute. Find T(160).
Answer: 80ºF |
| 4. Direct Variation |
| We can describe direct variation by using an equation of
the form y = kx |
7. The number of purchases n made at a toy store in a
mall varies directly with the number of people p entering the mall.
Write an equation describing this relationship if 75 purchases were made
on a weekday when 5,000 people mall visited the mall. Answer:
8. Use
your result from Exercise 7 to find the number of purchases the toy
store should make if an attendance of 9,000 is predicted for a special
Saturday promotion in the mall. Answer: 135 |
Graphing Using the Rectangular Coordinate System
| CONCEPTS |
| A rectangular coordinate system is composed of a
horizontal number line called the x-axis and a vertical number line
called the y-axis. |
1. a. Graph the points with coordinates (- 1, 3), (0,
1.5), (-4, -4), (2,7/2), and (4, 0). |
| The coordinates of the origin are (0, 0). |
2. Graph the following sets of coordinates
 |
| To graph ordered pairs means to locate their position on
a coordinate system |
3. In what quadrant does the point (-3, -4) lie? |
| Equations Containing Two Variables |
| An ordered pair is a solution if, after substituting the
values of the ordered pair for the variables in the equation, the result
is a true statement. |
4. Check to see whether (- 3, 5) i s a solution of y =
|2 + x | |
| Solutions of an equation can be shown in a table of
values. In an equation in x and y, x is called the independent
variable,
or input, and y is called the dependent variable, or output. To graph an
equation in two variables:
1. Make a table of values that contains several solutions written as
ordered pairs.
2. Plot each ordered pair.
3. Draw a line or smooth curve through the points. |
5. a. Complete the table of values
and the graph equation y = -x3
b. How would the graph of y = -x3 + 2 compare to the graph in part 6a. |
In many application problems, we encounter equations
that contain variables other than x and y.
 |
6. The graph below it shows the relationship between the
number of oranges “O” an acre of land will yield if “t” orange trees are
planted on it. a. If t = 70, what is “O”?
b. What importance does the point (40, 18) on the graph have? |
| Graphing Linear Equations |
| An equation whose graph is a straight line and whose
variables are raised to the first power is called a linear equation. |
7. Classify each equation as either linear or nonlinear.
 |
| The general or standard form of a linear equation is Ax
+ By = C where A, B, and C are real numbers and A and B are not both
zero. |
8. The equation 5x + 2y = 10 is in general form; what
are A, B, and C?
9. Complete the table of solutions for the equation 3x + 2y = - 18.
 |
To graph a linear equation:
1. Find three (x, y) pairs that satisfy the equation by picking three
arbitrary x-values and finding their corresponding y-values.
2. Plot each ordered pair.
3. Draw a line through the points. |
10. Solve the equation x + 2y = 6 for y, find three
solutions, and then graph it (see the table in 9 for help). |
| To find the y-intercept of a linear equation, substitute
zero for x in the equation of the line and solve for y. To find the
x-intercept of a linear equation, substitute zero for y in the equation
of the line and solve for x. |
11. Graph -4x + 2y = 8 by finding its x- and
y-intercepts. |
| Rate of Change and the Slope of a Line |
The slope m of a nonvertical
line is a number that measures
"steepness" by finding rise/run |
12. In each case, find the slope of the
line.
b. The line with the table of values shown here.
 |
 |
c. The line passing through the points (2, -5) and (5,
-5) |
If P(x1, y1) and Q(x2, y2) are, two points on a
nonvertical line, the slope m of line PQ is
 |
d. The line passing through the points –2 (1, -4) and
(3, -7) |
| Lines that rise from left to right have a positive
slope , and lines that fall from left to right have a negative slope, |
13. Graph the line that passes through (-2, 4) and has
slope m =-4/5 |
If a linear equation is written in slope-intercept form,
y = mx + b the graph of the equation is a line with slope m and
y-intercept (0, b). |
14. Find the slope and the y-intercept of each line.
15. Find the slope and the y-intercept of the line de termined by 9x - 3,
= 15. Then graph it. |
| Writing Linear Equations |
If a line with slope m passes
through the point (x1, y1), the .
equation of the line in
point-slope form is
 |
16 . Write the equation of a line
with the given slope that passes through the given point. Express the
result in slope-intercept form and graph the equation.
17. CAR REGISTRATION When it was 2 years old, the
annual registration fee for a Dodge Caravan was $380. When it was 4
years old, the registration fee dropped to $310. If the relationship is
linear, write an equation that gives the registration fee f in dollars
for the van when it is x years old. |
| Functions |
| A function is a rule that assigns to
each input value a single output value. |
18. In each case, tell whether a
function is defined.
a. y = 3x – 2
b. Is your age a function of your height? |
| For a function, the set of all
possible values of the independent variable x (the inputs) is called the
domain, and the set of all possible values of the dependent variable y
(the outputs) is called the range. |
19. Find the domain and range of the
function.
 |
| The notation y = f(x) denotes that y
is a function of x. |
20. For the function g(x) = 1 - 6x,
find each value.
 |
Quiz
The graph in Illustration 1 shows the number of dogs being
boarded in a kennel over a 3-day holiday weekend. Use the graph to answer
Problems 1-4.
1. How many dogs were in the kennel 2 days before the
holiday?
2. What is the maximum number of dogs that were boarded on
the holiday weekend?
3. When were there 30 dogs in the kennel?
4. What information does the y-intercept of the graph
give?
5. Graph y = x2 - 4
6. Is (-3, -4) a solution of 3x - 4y = 7?
7. Is y = x3 a
linear equation?
8. What are the x- and y-intercepts of the graph of 2x -
3y = 6?
9. Find the slope and the y-intercept of x + 2y = 8.
10. What is the slope of the line passing through (- 1, 3)
and (3, - 1)?
11. What is the slope of a line that is perpendicular to a
line with a slope -7/8
13. When graphed, are the lines y = 2x + 6 and 6x - 3y = 0
parallel, perpendicular, or neither?
14. Is this the graph of a function?
15. Does the equation y = 2x - 8 define a function?
16. If f(x) 2x - 7, find f(-3).
19. If g(s) = 3.5s3, find g(6). 9
Teamwork
DAILY HIGH TEMPERATURE For a 2-week period, plot the daily
high temperature for your city on a rectangular coordinate system. You can
normally find this information in a local newspaper. Label the x-axis
"observation day" and the y-axis "daily high temperature in degrees Fahrenheit."
For example, the ordered pair (3, 72) indicates that on day 3 of the observation
period, the high temperature was 72°F At the end of the 2-week period, see
whether any temperature trend is apparent from the graph.
MEASURING SLOPE Use a tape measure (and a level if
necessary) to find the slopes of five objects by finding rise/run. Record your
results in a chart like the one shown in Illustration 1. List the examples in
increasing order of magnitude, starting with the smallest slope.
Melissa,
TRANSLATIONS On a piece of graph paper, sketch the graph
of y = | x | with a black marker. Using a different color, sketch the graphs of
y = | x | +2 and y = | x | - 2 on the same coordinate system. On another pieceof
graph paper, do the same for y = | x | and y = | x + 2 | and y = |x – 2|. Make
some observations about how the graph of y = | x | is ,”moved” or "translated"
by the addition or subtraction of 2. Use what you have learned to discuss the
graphs of y = x2 y =x2 + 2, y = x2 - 2, y = (x + 2)2 , and y = (x - 2)2.
Melissa, has several books on her bookshelf, and if she
gets just a few more, she will have over 60 books. Exactly 20% of Melissa's
books are math books, and exactly 50% were Christmas presents. How many books
does she have on her shelf.
A man weighs 200 lbs plus one-fourth of his weight. How
much does he weigh?
An elephant and a little bird wish to play on a
teeter-totter. Let
E = the weight of the elephant.
b = the weight of the bird.
There must be some weight, w (probably very large), so
that
E = b + w.
Multiply both sides by E - b:
E(E - b) = (b + w)(E - b).
Using the distributive property:
E2 - Eb = bE + wE - b2 - wb.
Subtract ii-E from both sides:
E2 - Eb - wE = bE – b2 - wb.
Use the distributive property again :
E(E - b - w) = b(E - b - w).
Divide both sides by E – b - w
E = b.
Thus, the weights of the elephant and the bird are the
same and they would have no difficulty on the teeter-totter. Obviously, this
reasoning must be false but where is the error?
Graph each set on it’s own graph paper.
1. y = x, y = x + 1, y = x – 1
2. y = 2x, y = 2x + 1, y = 2x – 1
3. y = x/2, y = x/2 + 1, y = x/2 – 1
Discuss the similarities between the graphs and anything
else you notice.
In this exercise you will need to find 4 circular objects.
Use a string and determine the circumference and diameter of the object i.e. use
a pop can, table, coffee cup etc…
Make graph similar to the one below. Label your graph like
shown, put a scale on your graph too.
Determine your slope from two non-data points.
Helpful Hint: to determine the diameter place the objects
between books and measure edge-to-edge of the books.
What is the slope close to?
In your group discuss the graph in question 14 in your
quiz.
a. Is this a graph of a linear function
b. If so what is the slope of the function.
Solutions: Solve Linear Equations
3. quadrant III
4. not a solution
5. b. It would be 2 units higher
6. a. 9,000; b. It tells us 40 trees on an acre give the highest yield, 18,000
oranges.
7. a. nonlinear
8. A = 5, B = 2, C = 10
9.
10.
11. x-intercept: (-2, 0); y-intercept: (0, 4)
12. b. –7; c. 0; d. 2
13.
14. a. m = 3/4; y-intercept: (0, -2); b. m -4; y-intercept: (0, 0)
15. m = 3; y-intercept: (0, -5)
16. a. y = 3x + 2
17. f = -35x + 450
18. a. yes; b. no
19. a. D all reals; R: y ≤ 0;
20. a. –5; b. 37; c. –2; d. -8
