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Graphs : Inequalities

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Chapter 2: Graphing

Chapter 2 - GRAPHING

INEQUALITIES

Objectives

In this section, you will solve algebraic inequality problems and represent the answers on a number line. Also, you will identify quantities as being greater than, less than, or equal to each other.

The following example comes from Section 1.7 "Percentages," page 78.

Example 1. A math teacher, Dr. Pi, computes a student's grade for the course as follows:

20% for homework
50% for the average of 5 tests
30% for the final exam

Suppose Selena has an 89 homework average and a 97 test average. What does Selena have to get on the final exam to get a 90 for the course?

Let E be the variable that represents what Selena has to get on the final exam to get a 90 for the course.


Set up the equation.
90 = 66.3 + 0.30E
Simplified
23.7 = 0.30E
Subtracted 66.3 from both sides.
79 = E
Divided both sides by 0.30

Because Selena studied all semester, she only has to get a 79 on the final to get a 90 for the course.

A More Practical Version of the Problem

What Selena really wants for the course is an A*. So she would be happy with a grade of 95 or higher. The more realistic question would be:

What does Selena have to get on the final exam to get a 95 or higher for the course?

Let E be the variable that represents what Selena has to get on the final exam to get a 95 or higher for the course.

Explanation: We use the symbol ≥ to mean greater than or equal to. So instead of setting Selena's grade = to 95, we set Selena's grade ≥ 95.


Set up the equation.
66.3 + 0.30E ≥ 95
Simplified.
0.30 ≥ 28.75
Subtracted 66.3 from both sides.
E ≥ 95.67
Divided both sides by 0.30

Selena has to get a 95.67 or higher on the final to get a 95 or higher for the course.

The only difference between the two problems is using the signs = and ≥. Using ≥ makes the problem more realistic in terms of the grade Selena wants for the course.

Inequality Notation:

1. > means greater than.
For example, x > 3 represents all the numbers larger than 3, but not 3.

2. < means less than.
For example, x < 5 represents all the numbers smaller than 5, but not 5.

3. ≥ means greater than or equal to.
For example, x ≥ 7 represents all the numbers larger than 7, and includes 7.

4. ≤ +3 means less than or equal to.
For example, x ≤ 6 represents all the numbers smaller than 6, and includes 6.

Graphing Inequalities on the number line:
Graphing an inequality often conveys its meaning more clearly than just w*riting the inequality.

Rules: Two rules for graphing inequalities:

1. For ≥ and ≤ we use a shaded circle, ● , to show that we are including the number.

2. For > and < we use an opened circle, ○, to show that we are not including the number.

Example 2. Graph x ≤ -4 .

This means that we want the numbers smaller than and including -4. We put a shaded circle at -4 since we have ≤ and we draw an arrow going to the left of -4.

Example 3. Graph -5 < x ≤ 3.

This means that we want numbers between -5 and 3, including 3 but not including -5. We will put a shaded circle at 3 and an opened circle at -5 and draw a line between -5 and 3.

Solving Inequalities:
The only difference between solving equality equations and inequality equations is:

Rule: When you multiply or divide by a negative number, you must change the direction of the 3 inequality.

Explanation: Consider the two numbers 3 and 7. 3 < 7 because 3 is to the left of 7 on the number line.

If we multiply both numbers by a negative 1, then we get -3 and -7. Graphing -3 and -7 on the number line:

You can see that -7 is to the left of -3, so -3 > -7. Multiplying by a negative reverses the order of the numbers.

Example 4. Solve and graph the solution on the number line.

Subtracted -5 from both sides.
Divided both sides by -3. Since we divide both sides by a negative, we change the direction of the inequality.

Study Tip: The rule explaining the difference between solving equality equations and inequality equations should be written down on a note card and memorized along with an example demonstrating the rule.

Example 5. Solve and graph the solution on the number line.

Explanation: This problem has two inequalities. The solution should contain x in the middle. Whatever operations we do on one part we must do to all three parts.

Summary

Adding inequalities to your algebraic repertoire allows you to calculate when one quantity is more or less than another.

Major ideas:

1. When graphing inequalities on the number line, ≥, ≤ are represented by a small dot or shaded circle.
2. When graphing inequalities on the number line, <, > are represented by a small opened circle.
3. When solving a problem with more than one inequality:
   a. Perform the same algebraic step to all three parts of the inequality.
   b. The answer should have x in the middle.
4. When solving inequalities, if you multiply or divide by a negative number, then you must change the direction of the inequality.

APPLICATIONS OF INEQUALITIES

Objective

This section merges inequalities with the applications from the previous chapter.

Example 1. You are offered two sales positions after graduating from college. One, Math Inc. pays $10,000 plus 8% commission. The other, Hunter Company, pays $5,000 plus 12% commission. When does Math Inc. pay more than Hunter Company?

First determine the equations for each company. Create a table for each. Since both jobs are sales positions, we need a sales column, a calculation column, and a wage column. (You might be able to determine the equations by just reading the problem.)

The equation for Math Inc: w = 0.08S + 10,000

The equation for Hunter Company: W = 0.12S + 5,000

Now that the equations for both companies are determined, answer the question:

When does Math Inc. pay more than Hunter Company?

Set up the inequality. Subtracted 10,000 from both sides. Subtracted 0.12S from both sides. Divided both sides by -0.04 and change the direction of the inequality.

Math Inc. pays more than Hunter Company for sales less than $125,000.

Example 2. A phone company charges a basic minutes.

a. Find the equation for the cost of making phone calls over seven minutes. Simplify the equation.

Explanation: The company doesn't start charging a per minute rate until after the first seven minutes. You will have to subtract 7 from the number of minutes you were on the phone before multiplying by 0.08. This means you will have to use parentheses in the calculation column of the table.

The equation for the cost is: C = 0.08(m - 7) + 0.30


Simplified the equation.
C = 0.08m - 0.56 + 0.30
Use the Distributive Property.
C = 0.08m - 0.26
Combined Like terms.

Note that this formula is only valid when m > 7.

b. How many minutes were you on the phone if the cost was more than $3.00?


Cost is more than $3.00
0.08m - 0.26 > 3.00
Substituted cost equation for c.
0.08m > 3.26
Added 0.26 to both sides.
m > 40.75
Divided both sides by 0.08

If the cost is more than $3.00, then you were on the phone for more than 40.75 minutes.

c. How many minutes were you on the phone if the cost was between $2.50 and $3.25?


Cost is between $2.50 and $3.25.
2.50 < 0.08m - 0.26 < 3.25
Substituted cost equation for c.
2.76 < 0.08m < 3.51
Added 0.26 to all three parts.
34.50 < m < 43.9
Divided all three parts by 0.08

If the cost is between $2.50 and $3.25 then you were on the phone between 34.5 and 43.9 minutes.

Summary

This section allows us to expand the applications from the previous unit to problems that deal with quantities greater than, less than, or equal to each other. The procedures include the following steps:

1. Creating tables that include all necessary information.
2. Setting up the inequality.
3. Using algebraic skills to solve the inequality.

PLOTTING POINTS

Objective

This section describes the creation of graphs. A graph provides a visualization of the relationship between two quantities. A graph can be used to answer many questions.

Example 1. Identify the dependent and independent variables in the problem below.

Renting a van costs 10 cents per mile plus $20.00 a day.

The equation that relates miles and cost is c = 0.10m + 20.

The cost depends on the number of miles driven, so c is the dependent variable, and m is the independent variable.

Always write the ordered pair as (independent variable, dependent variable). The ordered pairs for this problem are (m, c).

The independent variable is always the horizontal axis.

The m (miles) axis is horizontal.

The dependent variable is always the vertical axis.

The c (cost) axis is vertical.

Example 2. When NASA sends a rocket into space, engineers monitor the temperature of certain gases. The table below gives a sample of the data collected. Note that negative time represents time before takeoff.

a. Create a graph based on the data.

Before we graph our data, we must decide:

What are the independent and dependent variables?

The temperature of the gas depends on how long the rocket is in the air.

Since the temperature depends on how long the rocket has been in the air, temperature is the dependent variable, so time is the independent variable. It is important to decide what the independent and dependent variables are because they determine the orientation of the graph.

In equations involving x and y, x is the independent variable, and y is the dependent variable.

How are ordered pairs of coordinates written?

In a non application problem, ordered pairs are the x and y coordinates written in parentheses and separated by a comma for example (x, y).

Always write the ordered pair as (independent variable, dependent variable).

In our example, (time, temperature).

The points we will graph are:
(-6, -27), (-4, -13), (-2, -4), (0, 4), (2, 43), (4, 21), (6, 9), and (8, -2).

What variable represents the horizontal axis?

The independent variable is always the horizontal axis.

In our example, time is the horizontal axis. In equations involving x and y, x is the horizontal axis.

What variable represents the vertical axis?

The dependent variable is always the vertical axis.

In our example, temperature is the vertical axis. In equations involving x and y, y is the vertical axis.

What scale should we use along the horizontal axis?

The scale is the distance between tick marks.

To decide the scale, find the lowest and highest value for time and think about the easiest way to count between them.

In our example, time ranges between -6 and 8, so we will count by twos. So the scale will be by 2.

What scale should we use along the vertical axis?

To decide the scale, find the lowest and highest value for temperature and think about the easiest way to count between them.

Since temperature ranges between -27 and 43, we will count by fives, starting at -30 and going to 45. The scale will be 5.

The graph of the data is given below.

b. Answer the following questions based on your graph.

1. Estimate the temperature three minutes into the flight.

Three minutes is three units to the right on the horizontal axis, Time. If you lightly draw a vertical line up from 3, the point where the vertical line intersects the graph is the answer. Put a dot at this point and read the temperature on the vertical axis. The temperature of the gas at three minutes is approximately 35 degrees Celsius.

2. What was the temperature at takeoff?

When the rocket takes off, the time is zero. The point that represents the temperature of the gas is the point on the Temperature axis. This point is called the Temperature intercept because it is on the Temperature axis. Put a dot at this point and read the temperature on the vertical axis. The temperature at takeoff is 4 degrees Celsius.

3. When was the temperature zero?

The temperature is zero when the graph crosses the Time axis. These points are called the Time intercepts because they are on the Time axis. Put a dot at these points and read the time on the horizontal axis. The temperature is 0 just before takeoff, approximately -0.5, and about 7.5 minutes.

4. When did the temperature increase the fastest?

The temperature increased the fastest between 0 minutes and 2 minutes. This is where the graph is steepest.

Summary

This section covers the basic process of graphing. You must master these concepts to be successful in basic algebra.

Vocabulary:
1. Independent Variable.

The independent variable was the first column in the tables made in previous sections. Miles, time, and sales are usually independent variables.

2. Dependent Variable.

The dependent variable is the quantity that is contingent on the independent variable. The dependent variable was the third column in the tables made in previous sections. Cost and wages are usually dependent variables.

3. Ordered Pair.

The ordered pair indicates the coordinates of a point on the graph. It always has the form: (Independent Variable, Dependent Variable) or (x, y).

4. Scale.

The scale is the distance between the tick marks on the axis.

5. Intercept.

The intercept is the place where the graph crosses an axis.

    The graph below highlights the 5 definitions:
  • The x axis is the horizontal axis, and x is the independent variable.
  • The scale of the x axis is 5.
  • The y axis is the vertical axis, and y is the dependent variable.
  • The scale of the y axis is 25.
  • The ordered pair is indicated by (x, y).
  • The x intercepts are approximately (-2.5, 0) and (27.5, 0).
  • The y intercept is approximately (0, 75).

6. This type of graph uses the Cartesian coordinate system.

INTERPRETING GRAPHS

Objective

This section covers the basic properties of graphs. You will learn about intercepts, vertices, and intersections. In order to answer questions correctly, pay attention to the scale of the graph.

Example The graph below shows the profit of two toy companies, Radio Control Inc. and Turbo Car Co.

Label the variables:

Let N represent the number of toys sold. The units for N are millions.

Let P represent the profit of the companies. The units for P are thousands.

1. What is the independent variable?

The independent variable is the horizontal axis, which in this example is the number of toy cars sold, N.

2. What is the dependent variable?

The dependent variable is the vertical axis, which in this case is profit, P.

3. How many toy cars does Radio Control Inc. have to sell to break even?

A company breaks even when the profit is zero. So the break even points are on the horizontal axis. The graph of Radio Control Inc. crosses the N axis at approximately (1.1, 0) and (6.6, 0). The break-even points are the N intercepts. Radio Control Inc. breaks even when sales are 1,100,000 and 6,600,000 toy cars.

4. For what number of cars sold is the profit the same for both companies?

The two companies have the same profit where the two graphs intersect or cross. Put a dot where the two graphs intersect. The answer to the question is the independent variable, number of cars sold, N. The values for the independent variable are approximately 1.6, and 5.9. The two companies make the same profit when they sell 1,600,000 or 5,900,000 toy cars.

5. How much money does Turbo Car Co. lose if no cars are sold?

If Turbo Car Co. doesn't sell any cars, then the independent variable, N, is zero. So the answer to the question is the Profit intercept. The profit coordinate of the Profit intercept is -5. Turbo Co. loses $5,000 if no toy cars are sold.

6. For what number of cars does Radio Control Inc have a maximum profit?

The maximum profit is at the top of the graph. This point is called the vertex of the graph. The N (number of cars sold) coordinate is approximately 4. Radio Control Inc. will have a maximum profit if 4,000,000 cars are sold.

7. What is the maximum profit of Radio Control Inc.?

The maximum profit occurs when Radio Control Inc. sells 4 million toy cars, N = 4. At that point, their profit will be $14,000.

Summary

Interpreting graphs is an extremely important skill. Visual representation of data is expected by both supervisors and customers. You should be comfortable with graphing concepts.

Vocabulary:
1. The Cartesian coordinate system is comprised of a horizontal axis and a vertical axis.

2. The independent variable was the first column in the tables made in previous sections. Miles, time, and sales are examples of independent variables. The independent variable is represented by the horizontal axis.

3. The dependent variable is the quantity that is contingent on the independent variable. The dependent variable was the third column in the tables fromprevious sections. Cost and wages are examples of dependent variables. The dependent variable is represented by the vertical axis.

4. A point on the Cartesian coordinate system is called an Ordered Pair. An ordered pair always has the form (independent variable, dependent variable).

5. The numbers in an ordered pair are called coordinates.

6. The intercepts are the points where the graph crosses an axis. One of the coordinates of an ordered pair representing an intercept is always zero.

7. The intersection is the point where two graphs cross.

8. The vertex is the peak or bottom of a curve.

Study Tip: Many students confuse intercepts and intersection. Draw a graph on a note card and label all the intercepts and intersections.

GRAPHING LINES BY PLOTTING POINTS

Objective

In this section, we will graph lines by finding three ordered pairs or points. Two points determine the line; the third point is used as a check that the other two are correct.

Example 1. You need to rent a moving van. Class Movers charges a basic rate of $24.95 plus $0.32 per mile.

(This example is from the Section 1.2 Introduction to Variables, page 20 and Section 1.3 "Solving Equations, page 44.)

a. Calculate the cost of renting a van if you drive the following number of miles.

The cost equation is. c = 0.32*m + 24.95

b. What are the independent and dependent variables?

The cost depends on the number of miles driven. So cost is the dependent variable, and miles are the independent variable.

c. What are the ordered pairs generated by the table?

This example has the form (miles, cost) because ordered pairs have the form (independent variable, dependent variable).

The ordered pairs are (10, 28.15), (20, 31.35), (30, 34.55).

d. Graph the equation c = 0.32m + 24.95.

Example 2. Graph 2x + 5y = 26

  • x is the independent variable
  • y is the dependent variable.
  • You must have two points or ordered pairs to graph a line, but we will find three. We use the third point to check our work. All three points must be on the line.

Choose three values for x; then use algebra to find y.

Explanation: It doesn't matter what 3 values you pick for x, but don't choose numbers too close together and chose a negative number.

Graph the ordered pairs (-5, 7.2), (2, 4.4), and (15, -0.8)

Explanation: The equation 2x + 5y = 26 contains two variables. A solution to this equation must include a number for x and a number for y. The point (2, 4.4) is a solution to this equation. The line represents all of the solutions to the equation.

You need to choose three values for x. Since x is multiplied by the fraction , picking multiples of three will make the arithmetic easier, even if you use a calculator. Once you pick a value for the independent variable x, substitute it into to find the dependent variable y.

1. Choose x = -6 and substitute into

Substituted x = -6 into the equation.
y = 4(-2) - 8
Divided -6 by 3.
y = - 8 - 8
Multiplied 4 and -2.
y = - 16.
Subtracted 8 from -8.

The ordered pair is (-6,16).

1. Choose x = 3 and substitute into

Substituted x = 3 into the equation.
y = 4(1) - 8
Divided 3 by 3.
y = 4 - 8
Multiplied 4 and 1.
y = - 4.
Subtracted 8 from 4.

The ordered pair is (3,-4).

1. Choose x = 9 and substitute into

Substituted x = 9 into the equation.
y = 4(3) - 8
Divided 9 by 3.
y = 12 - 8
Multiplied 4 and 3.
y = 4.
Subtracted 8 from 12.

The ordered pair is (9,4).

Explanation: What happens if you pick a value for x that is not a multiple of 3? For example, choose x = 5,

The arithmetic is harder.

Graph the three ordered pairs or points (-6, -16), (3, -4) and (9, 4).

Explanation: The purpose of this example is to convince you that if you take a moment to carefully choose the values for x, then the arithmetic of the problem can be a lot easier.

Summary

Graphing is a very important topic in algebra. A graph provides visualization of data or of an equation. For example, a graph allows you to see how the cost of renting a van is increasing as you drive more miles.

To graph a line:

  1. Choose three different values to substitute for the independent variable. A minimum of two points is needed to graph a line. A third point is used to check the work.
  2. For each of your chosen values of the independent variable, use algebra to find the corresponding value of the dependent variable.
  3. Plot each point and connect them using a ruler. If the points are not forming a line, then you have made a mistake.
  4. In a nonapplication problem, x is always the independent variable, and y is always the dependent variable.
  5. In a nonapplication problem, there are no restrictions on the variables. Your graph should include both positive and negative values for x and y.
  6. When graphing an application, you need to think about the restrictions on the variables. In Example 1 on page 140, it wouldn't make any sense to have negative values for miles.

Study Tip: Note card for graphing

  1. Construct a table of values by choosing values for x; then calculate the corresponding value for y.
  2. Use graph paper, label axis, and set the scale.
  3. Plot the points.
  4. Connect the points.

Review the card as homework and use the card as a reference when you do the homework from this section.

GRAPHING LINES BY PLOTTING INTERCEPTS

Objectives

This section presents an additional way to graph a line. To graph a line, you need a minimum of two points. Two special points can be used. They are the intercepts of each axis. Often the intercepts have special meanings in a mathematical model. Also covered in this section are horizontal and vertical lines.

Example 1. An 8-year-old boy plans to open a lemonade stand. All of his supplies cost $18.00, and he charges $0.50 per glass. The equation that relates profit and number of glasses sold is

P = 0.50g-18.00

a. What are the independent and dependent variables?

His profit depends on the number of glasses sold. So P is the dependent variable, and g is the independent variable. The points on the graph will have the form (g, P).

b. How many glasses does he have to sell to break even?

Breaking even means that his profit will be zero.

Find g when P = 0.

Substituted 0 for P.
18.00 = 0.50g
Added 18.00 to both sides.
36 = g.
Divided both sides by 0.50.

He needs to sell 36 glasses to break even. Graph the point (36, 0).

c. How much money will he make if he doesn't sell any lemonade?

If he doesn't sell any, then g = 0. Find P when g = 0.


Substituted 0 for g.
P = -18.00
Computed P.

He will lose $18.00 if he doesn't sell any glasses of lemonade. Graph the point (0, -18).

Explanation: Parts b and c are illustrated in the table below.

d. Graph the line P = 0.50g -18.00 by plotting the points obtained in Parts b and c.

Choose an appropriate scale and only graph the portion which makes sense in the problem. Label the axes.

Vocabulary: : The point (36, 0) is g intercept because the point is on the g axis.

The point (0, -18) is the P intercept because the point is on the P axis.

The significance of the previous example is:

The point (36, 0) is the g intercept. The P coordinate of the point (36, 0) is zero.

The point (0, -18) is the P intercept. The g coordinate of the point (0, -18) is zero.

To find the intercept of one of the axes, set the other variable equal to zero.

Example 2. Graph the line 25x + 0.04y = 50 by finding the intercepts.

a. Find the y intercept. Set x = 0.

+ 0.04 y = 50
Substituted x = 0.
0.04y = 50.
Multiplied = 0
y = 1250
Divided both sides by 25.

The y intercept is (0, 1250).

b. Find the x intercept. Set y = 0.

25x + = 50
Substituted y=0.
25x = 50
Multiplied = 0.
x = 2.
Divided both sides by 25.

The x intercept is (2, 0).

c. Graph the points (0, 1250) and (2, 0).

You have to think about the scale of the y axis. The scale of the y axis is 500. It is nice, but not necessary, to have the intercept correspond to an interval on the axis. Also, you don't want the intercept to be at the very top or bottom of the graph. Other possibilities exist. Also, the scale of the x axis is different than the scale of the y axis.

Study Tip:

Write a note card describing the process of graphing by finding the intercepts. Review regularly.

Horizontal Lines:

Example 3. Today we will look at the cost of renting a car. AUTO will rent it to us for a flat daily rate of $55 with no mileage charge. (This is from the group work on page 27.)

The equation is: AUTO: CA = 55.00

The purpose of this example is to graph the cost equation for AUTO.
The company AUTO charges $55.00 no matter how many miles you go.

If you drive, 10 miles it costs $55. The ordered pair is (10, 55).
If you drive, 90 miles it costs $55. The ordered pair is (90, 55).

The graph of C = 55.00 is indicated below.

Vocabulary: The graph is horizontal or flat because the cost never changes.
The equation of a horizontal line is y equal to a constant, y = c.

Example 4. Graph y = -4.

Explanation: Every point on the graph has a y coordinate of - 4.
Points on the graph: (-6,-4), (0,-4), (3,-4).

Vocabulary: The equation of a vertical line is the independent variable equal to a constant. The graph of x = h, h a constant, is a vertical line.

Example 5. Graph x = 3.

Explanation: Every point on the graph has an x coordinate of 3.
Points on the graph: (3, -8), (3, 0), (3, 6).

Study Tip:

You should write the equations for horizontal and vertical lines on a note card. You will need to graph horizontal lines in the Section 2.9 Applications of Graphs. You should review this card at least twice a week.

Summary

Graphs allow you to visualize the equation. You now know two ways to graph a line, plotting any three points or finding the intercepts. Sometimes you will have to decide which way is easier.

A. To graph a line by plotting three points.

  1. Choose two values of the independent variable.
  2. For each value of the independent variable use algebra to find the value of the dependent variable.
  3. Plot and connect the two points.
  4. Choose a third value of the independent variable to check your work.

B. Graphing lines by finding the intercepts.

  1. To find the x intercept, set y = 0 and solve for x.
  2. To find the y intercept, set x = 0 and solve for y.

C. Horizontal lines.

  1. The equation of a horizontal line is: y = a number
  2. The graph of a horizontal line:

D. Vertical lines.

  1. The equation of a vertical line is: x = a number
  2. The graph of a vertical line is:

INTRODUCTION TO SLOPE

Objectives

This section introduces the important concept of slope using applications from previous sections. Slope describes the rate at which a line either rises or falls.

Example 1. You need to rent a moving van. Class Movers charges a basic rate of $24.95 plus $0.32 cents per mile.
(This example is from Graphing Lines by Plotting Points, page 140.)

a. Calculate the cost of renting a van if you drive the following number of miles.

b. The graph of the equation c = 0.32m + 24.95 is presented below.

c. The slope of this line is computed by selecting any two points to find

Explanation: 34.55 - 28.15, represents the change in cost. Cost is the dependent variable and the vertical axis.
30-10, represents the change in miles. Miles are the independent variable and the horizontal axis.

d. Compute and interpret what it means.

0.32 is the cost per mile and is the coefficient of m, the independent variable. This is not a coincidence. In this problem, slope is the cost per mile.

e. Suppose Class Movers begins to charge 55 cents per mile. Then the new equation is c = 0.55m + 24.95. The slope is still the cost per mile, so the slope of this line is 0.55. The graphs of both equations appear below.

The line representing the equation c = 0.55m + 24.95 is steeper than the line representing c=0.32m-24.95 because Class Movers charge more per mile. The slope of the line c = 0.55m + 24.95 is greater than the slope of the line c = 0.32m + 24.95.

Slope measures the steepness of a line.

Example 2. Which plant grows faster: Hybrid A sunflower which grew 26 inches in 10 days or Hybrid B sunflower which grew 28 inches in 9 days?

Hybrid B grew faster than Hybrid A. Since plants grow at different rates, the numbers 2.6 and 3.111 represent the average growth per day. The numbers 2.6 and 3.111 also represent the idea of slope,

Slope also means the average rate of change. The slope of Hybrid B is steeper than the slope of Hybrid A.

Example 3. The average cost of a personal computer is shown in the table below.

a. Make a graph of year versus cost.

b. Calculate the average rate of change.

On the average, the cost of personal computers decreased $43 per year between 1996 and 2010.

Explanation: Look at the graph above:
1,076 - 1,678 represents the vertical change.
2001 -1987 represents the horizontal change.

If the slope of the line is negative, then the line is decreasing (moving down left to right).

Conversely, if the slope is positive then the line is increasing (moving up left to right).

Example 4. Use the data from Example 3 to calculate the percent change in the cost of personal computers.

Percent change is a similar idea to slope. It also measures how a quantity changes; however, percent change only involves one variable. In our problem, the change involves only the cost.

The formula for percent change is:

The variable New is the cost of personal computers in 2001, $1,076.

The variable Old is the cost of personal computers in 1987, $1,678.

Explanation: Notice the numerator of the formula for percent change is the same as the numerator for the formula for slope.

The cost of personal computers decreased 35.88% from 1987 to 2001.
From example 3, the slope is -43; the cost of personnel computers decreased on average $43 per year between 1987 and 2001.

Summary

Slope is a very important topic in algebra. It measures how things change. The following are different interpretations of slope:

Slope

  • Slope measures the steepness of a line.
  • Slope is the average rate of change.
  • If the slope of the line is negative, then the line is decreasing (moving down left to right).
  • If the slope of the line is positive, then the line is increasing (moving up left to right.)

Percent Change also measures changes in a quantity. The formula for Percent Change is:

Study Tip: You should write this formula on a note card and memorize it.

SLOPE

Objective

This section will cover the algebraic formula for slope, including the slopes of horizontal and vertical lines, and the slope-intercept equation of a line.

Algebraic formula for slope:

Let (x1, y1) and (x2, y2) be any two points on the line; then the formula for slope is:

Study Tip: Write the formula on a note card for easy reference.

Example 1. Graph the line that passes through the two points (-2, -1), (3, 5) and find the slope.

a. To graph the line, just plot the two points.

b. Use the formula to find the slope.

Explanation: It doesn't matter which y value you choose to equal y1 as long as you are consistent. If y1 = 5, then x1 has to be 3. (3, 5) is a point on the graph.

Since the slope is positive, the line is increasing or rising from left to right.

Example 2. Given the graph below, find the slope.

Since the slope is negative, the line is decreasing or falling from left to right.

Example 3. Graph the line that passes through the two points (-3, 4), (5, 4) and find the slope.

a. To graph the line, just plot the two points.

b. Use the formula to find the slope.

The slope of a horizontal line is zero.

Explanation: The slope of a horizontal line is zero because there is no vertical change.

Study Tip: This is an important fact that will be significant in a later section.

Example 4. Graph the line that passes through the two points (3, -4), (3, 2) and find the slope.

a. To graph the line, just plot the two points.

b. Use the formula to find the slope.

The slope of a vertical line is undefined since division by zero is undefined.

The Slope-Intercept Equation of a Line

Vocabulary: : The equation y = mx + b is called the slope-intercept equation of a line. The coefficient of x, m, is the slope and (0, b) is the y intercept. The equation is called a first degree polynomial because x is raised to the first power.

All of the applications in this section are in the form of the slope-intercept equation.

Example 5. You need to rent a moving van. Class Movers charges a basic rate of $24.95 plus 32 cents per mile.

This is the basic problem for the first half of the course, and we know the equation relating cost and miles is:

c = 0.32m + 24.95

The slope of the line is 0.32 since 0.32 is the coefficient of variable m, miles.

The c intercept is (0, 24.95).

Explanation: The letter m is used differently in the definition than it is used in the application. In the definition, m represents slope. In the example, m is the variable for the number of miles.

Study Tip: y = mx + b is an important equation you need to know. Write it on an index card for further reference. Know that m is the slope and (0, b) is the y intercept.

Summary

Slope is a fundamental concept in mathematics. It measures how things change.

The basic ideas in this section:

  1. The formula for the slope of a line is .
  2. If the slope of a line is positive, then the line is increasing or rising from left to right.
  3. If the slope of a line is negative, then the line is decreasing or falling from left to right.
  4. The slope of a horizontal line is zero.
  5. The slope of a vertical line is undefined.
  6. In the slope-intercept equation, y = mx + b, m is the slope, and (0,b) is the y intercept.

APPLICATION OF GRAPHS

Objectives

This section summarizes the course thus far. You will create tables to find equations, apply your knowledge of graphing lines by finding the intercepts and plotting points, interpret slope, understand inequalities graphically, and solve equations.

Vocabulary: Review

  • Tables: A systematic arrangement of information using rows and columns. (Section 1.2 "Introduction to Variables)
  • Solving Equations: Algebraic technique used to determine the point when two quantities are equal. (Section 1.4 Solving Equations)
  • Inequalities: The use of less than (<) and greater than (>) symbols to show relationships. (Section 2.2 Applications of Inequalities)
  • Plotting Points: Using numbers generated by the table to graph the line. (Section 2.5: Graphing Lines by Plotting Points)
  • Intercept: The point where the graph crosses either the x or y axis. (Section 2.6: Graphing Lines by Plotting Intercepts)
  • Slope: (Steepness of a line.) Change in the dependent variable divided by change in the independent variable. Often slope is described by cost per mile or something similar. (Section 2.7 Introduction to Slope)

Example. You are going to rent a car for a day. You have two choices, Speed Car Rental and Honest Car Rental. Speed charges $18 plus $0.85 per mile while Honest charges $42 plus $0.45 per mile.

a. Develop an equation for the cost of renting a car from Speed.

The equation for Speed is c = 0.85m + 18.

Suggestion: By now, you may be able to derive the cost equation by reading the problem. Using descriptive variable names like m for the number of miles driven and c for cost should help you interpret what the formula means.

Develop an equation for the cost of renting a car from Honest.

The equation for Honest is c =0.45m + 42.

Suggestion: Instead of making a table, you could have recognized that the cost per mile is multiplied by the variable m, and the flat cost is the constant.

c. Find the intersection of two lines. Label the point.

Study Tip: Question c is similar to calculating how many miles you have to drive for the two companies to charge the same. This was covered in Section 1.5 "Applications of Linear Equations.

Step 1. Find the number of miles that result in the same cost.

Set the cost equations equal to each other. Subtracted 0.45m from both sides. Subtracted 18 from both sides. Divided both sides by 0.40

The two companies charge the same when you drive 60 miles.

Step 2. Find the cost of going 60 miles

Choose one of the equations and plug in 60 for m.

Speed:

C = .85 ● 60 +18

C = 69

The two lines intersect at (60, 69).

Explanation: It doesn't matter which company you choose to substitute 60 for miles; you will get the same cost.

d. Graph both equations on the same set of axes. Label each axis and choose an appropriate scale. Only graph the portion that is relevant to the problem.

Step 1. We need to decide which variable is the independent and which is the dependent.

Since cost depends on miles, cost is the dependent variable, and miles are the independent variable. So we will write our points as (miles, cost), and our graph will be:

Step 2. Find at least two points for each line, Speed and Honest.

Point 1. From Part c, we know that the lines intersect at (60, 69). This point will be used to graph both lines.

Point 2. Find the cost intercept for Speed and Honest. We find the cost intercept by letting m = 0.

Step 3. Plot the points and label the graph.

Study Tips

  1. Organize your work in tables.
  2. Note that the intersection and the intercept for each company are two different points.
  3. You should use graph paper and a ruler when you make a graph.
  4. You should use different colored pencils, and you should label each line, the intersection, and the cost intercepts.

e. Use the graph to determine when Speed costs more than Honest.

Speed is more expensive than Honest when the graph of Speed is above the graph of Honest. This is when m is greater than 60, ( m > 60). So Speed costs more than Honest when miles are greater than 60.

f. Use the graph to determine when Honest costs more than Speed.

Honest is more expensive than Speed when the graph of Honest is above the graph of Speed. This is when m is less than 60, (0 < m < 60). So Honest costs more than Speed between 0 and 60 miles.

g. What do the cost intercepts mean in terms of the problems?

The cost intercept of Speed is where the line for Speed crosses the Cost axis. This is the point (0, 18). The cost of going zero miles is $18.

The cost intercept of Honest is where the line for Honest crosses the Cost axis. This is the point (0, 42). The cost of going zero miles is $42.

h. What does the slope of each line mean in terms of the problem?

We can find the slope of each line by using the slope-intercept equation

y = mx + b.

The number multiplying x (the coefficient) is the slope of the line.

For Speed, c = 0.85m + 18, the slope of the line is 0.85.
For Honest, c = 0.40 m + 42, the slope of the line is 0.40.

In both cases, the slope of the line is the cost per mile and indicates the steepness of the line and the rate the cost is increasing per mile.

Summary

This section reviews most of the material presented thus far in the course. You should be able to look at the equation

c = 0.85m + 18

and understand:

a. 0.85 is the slope of the line, measures its steepness, and the cost per mile.

b. 18 is the cost intercept, and it represents the cost of going 0 miles.

c. Intersection and intercept are two different terms.

The intersection is where two lines cross. The intersection is found by setting the two equations equal to each other and solving algebraically for x (or the independent variable). Then the y coordinate (or dependent variable) is found by substituting the solution into one of the equations.

An intercept is where the line touches one of the axes. The x intercept of an equation is found by setting y = 0. The y intercept is found by setting x = 0.

Study Tip: Write the definitions of intercept and intersection along with a graph on the same note card so that you realize that they are different.

CHAPTER 2 REVIEW

This unit interprets algebra and inequalities graphically. You should be able to read a problem and construct a graph that displays all of the important features of the problem.

Section 2.1 Inequalities:

Solving equations of inequalities is similar to solving traditional algebraic equations except, when you multiply or divide by a negative, you must change the direction of the inequality symbol.

Some inequality equations have three parts. The variable is to be isolated in the middle.

When graphing inequalities on the number line, use a shaded circle, ● ,for ≤ or ≥ and an opened circle, o, for < or >.

Section 2.2 Applications of Inequalities:

Inequalities interpret phrases like more than and less than in mathematical models studied in the previous unit.

Example 4. The equation C=0.12m+0.40 represents the cost of making a long distance phone call. M is the number of minutes on the phone. If the cost was more than $1.25 and less than $1.40, how long were you on the phone?

If the phone call cost between $1.25 and $1.40, then you were on the phone between 7.08 minutes and 8.333 minutes.

Section 2.3 Plotting Plots

Important Vocabulary words:

The independent variable is the one represented by the first column of a table and is the horizontal axis. In equations involving x and y, x is the independent variable.

The dependent variable is the one represented by the last column of the tables and is the vertical axis. In equations involving x and y, y is the dependent variable.

Since there are two variables on our graph, we must be consistent in how we describe a point on the graph. An ordered pair describes this point. It always has the form (independent variable, dependent variable) or in a nonapplication problem (x, y).

Example 5. The equation c = 0.12m + 0.40 represents the cost of making a long distance phone call. M is the number of minutes on the phone.

Construct a table.

Since the cost of making a phone call depends on how long you were on the phone, c is the dependent variable, and m is the independent variable.

An ordered pair will have the form (m, c).

The graph has the form

Section 2.4 Interpreting Graphs:

The important features of a graph are:

  • Vertex: The high or low point.
  • Intercept: The point at which the graph crosses the horizontal or vertical axis.
  • Intersection: The point where two graphs meet.
  • Slope: The steepness of a line and the average rate of change.

Section 2.5 Graphing Lines By Plotting Points:

Use the points from a table to generate a graph.

Example 6. A math textbook company, Calculate Inc., offers you a job selling textbooks. They pay $15,000 plus 9% commission. Complete the table below and make a graph of your possible wages.

Graph the points (50,000, 19,500), (100,000, 24,000), and (200,000, 33,000).

Choosing the scale of the graph:

For the Sales axis:
Start at zero and go past $200,000-perhaps $250,000. Count by ten thousands.

For the Wages axes:
Start at zero and go past $33,000--perhaps $40,000. Count by five thousands.

Section 2.6 Graphing Lines by Finding Their Intercepts:

To find the x intercept, set y = 0.

To find the y intercept, set x = 0.


Section 2.7 Introduction to Slope:

  • Slope measures the steepness of a line.
  • Slope is the average rate of change.
  • If the slope of the line is negative, then the line is decreasing left to right.
  • If the slope of the line is positive, then the line is increasing left to right.

Percent Change also measures how a quantity changes. The formula for Percent Change is:

Example 8. Use the information in the table below to answer the questions.

a. Find the average rate of change.
(This is the same as the slope of a line.)

Year is the independent variable, and number of stations is the dependent variable.

On the average, there was a decrease of 12 radio stations per year.

b. Find the percent change.

The number of jazz radio stations decreased by 59.67% between 1999 and 2010.

Section 2.8 Slope

The algebraic formula for slope is m =

The slope of a horizontal line is zero.

The slope of a vertical line is undefined.

The slope-intercept equation is y = mx + b.

Example 9. Find the slope of the line that contains the points (-3, 5) and (2, -1).

Section 2.9 Application of Graphs

This section summarizes the major concepts of the course thus far.

Example 10. You need to rent a moving van. One company, Quick Movers, charges a basic rate of $24.95 plus $0.32 a mile. A second company, Silver Glove Movers, charges a basic rate of $19.95 plus $0.40 a mile.

The equations are:
Quick: c = 0.32m + 24.95
Silver: c = 0.40m + 19.95

To find where the two lines intersect:
0.40m + 19.95 = 0.32m + 24.95
0.40m = 0.32m + 5.00
0.08m = 5.00
m = 62.5

Find C when m = 62.5.
c = 0.32.62.5 + 24.95
c = 44.95

The two lines intersect at (62.5, 44.95)

Explanation: You can use either company's equation.

Graph the two equations.

Silver Co. costs more if you drive over 62.5 miles.
Quick costs more if you drive less than 62.5 miles.

The cost intercept of Quick is (0, 24.95). It will cost you $24.95 to drive nowhere.

The slope of c = 0.40m + 19.95 is 0.40. Silver company charges 40 cents per mile.

Study Tips:

  1. Make sure you have done all of the homework exercises.
  2. Practice the review test starting on the next page by placing yourself under realistic exam conditions.
  3. Find a quiet place and use a timer to simulate the class length.
  4. Write your answers in your homework notebook or make a copy of the test. You may then re-take the exam for extra practice.
  5. Check your answers. The answers to the review test are on page 451.
  6. There is an additional exam available on the Beginning Algebra web page, see page xi.
  7. Review often. Do NOT wait until the night before to study.