lesson 8: graphs and graphing linear equations...2014/06/09 · lesson 8: graphs and graphing...

49

Lesson 8: Graphs and Graphing Linear Equations

A critical skill required for the study of algebra is the ability to construct and interpret graphs. In

this lesson we will learn how the Cartesian plane is used for constructing graphs and plotting

data. We will interpret the points and behavior of a graph with respect to the input and output

variables. We will also learn the rules/guidelines for constructing good graphs so that the graphs

we create can be easily read and understood.

Lesson Topics

Section 8.1: The Cartesian plane

Input and Output Variables

Ordered pairs

Plotting points

Quadrants

Section 8.2: Constructing good graphs from Data

Section 8.3: Linear Equations – Two Variables

Section 8.4: Graphing linear equations by plotting points

Section 8.5: Intercepts

Section 8.6: Horizontal and Vertical Lines

Lesson 8 Notes

Name: ________________________________ Date: _____________

Page 151

Mini-Lesson 8

Section 8.1: The Cartesian Plane

In this chapter, we will begin looking at the relationships between two variables. Typically one

variable is considered to be the INPUT, and the other is called the OUTPUT. The input is the

value that is considered first, and the output is the value that corresponds to or is matched with the

input. The input/output designation may represent a cause/effect relationship, but that is not

always the case.

Ordered Pairs

Example 1: Ordered Pairs (input value, corresponding output value)

Input Output Ordered Pairs (input, output)

4 –3

5 8

(0, –4)

(–2, 6)

Example 2: The Rectangular Coordinate System (Cartesian Coordinate System)

Lesson 8: Graphs and Graphing Linear Equations Mini-Lesson

Page 152

Plot and label the points.

A. (–4, 2)

B. (3, 8)

C. (0, –5)

D. (–6, –4)

E. (5, 0)

F. (2, –8)

G. (0, 0)

Quadrants

Quadrant Coordinates

I (+, +)

II (–, +)

III (–, –)

IV (+, –)

You Try

1.

Plot and label the points.

A. (6, –3)

B. (1, 9)

C. (–4, 0)

D. (–2, –8)

E. (0, 5)


Page 153

Section 8.2: Constructing a Graph from Data

Criteria for a Good Graph

1. The horizontal axis should be properly labeled with the name and units of the input variable.

2. The vertical axis should be properly labeled with the name and units of the output variable.

3. Use an appropriate scale.

Start at or just below the lowest value.

End at or just above the highest value.

Scale the graph so the adjacent tick marks are equal distance apart.

Use numbers that make sense for the given data set.

The axes meet at (0,0) Use a “//” between the origin and the first tick mark if the scale

does not begin at 0.

4. All points should be plotted correctly, and the graph should make use of the available space.

Example 1: The table below shows the total distance (including reaction time and

deceleration time) it takes a car traveling at various speeds to come to a complete stop.

Speed (miles per hour) 15 25 35 45 50 60 75 80

Stopping Distance (ft) 44 85 135 196 229 304 433 481


Page 154

You Try

2. Consider the following data set.

Elapsed time (seconds) 0 1 1.5 2.4 3 3.8

Height of Golf Ball (feet) 0 59 77 88 81 54

a. What is the input variable? ________________________________________

b. What was the height of the ball after 3 seconds?_____________________________

c. After how many seconds was the ball 77 feet in the air? _______________________

d. In a complete sentence, interpret the meaning of the ordered pair (1, 59).

e. Construct a good graph of this data.


Page 155

Section 8.3: Linear Equations - Two Variables

Linear Equations

Example 1: Verify that the ordered pairs below satisfy the equation y = 2x + 3.

(–2, –1) (0, 3) (1, 5)


Page 156

Example 2: Verify that the ordered pairs below satisfy the equation 3x + 2y = 6.

(–2, 6) (0, 3) (2, 0)

You Try

3. Verify that the ordered pairs below satisfy the equation y = 3x – 2.

(–2, –8) (3, 7) (0, –2)


Page 157

Section 8.4: Graphing Linear Equations - Two Variables

Graphing Linear Equations by Plotting Points

Step 1: Choose two or more values for the input variable, and list them in a table of values.

Step 2: Substitute each input value into the equation and compute the corresponding output

values. List these values in the table.

Step 3: Write each input-output pair in the table as an ordered pair.

Step 4: Plot the ordered pairs, connect them, and extend the line beyond the points to show that

the pattern continues.

Example 1: Graph the equation y = 3x – 2

x y Ordered Pair

Example 2: Graph the equation

x y Ordered Pair


Page 158

Example 3: Graph the equation 4x + 2y = 10

x y Ordered Pair

You Try

4. Graph the equation

x y Ordered Pair


Page 159

Section 8.5: Intercepts

Vertical and Horizontal Intercepts

The vertical intercept (y-intercept) is the point at which the graph crosses the vertical axis.

The input value of the vertical intercept is always____________

The coordinates of the vertical intercept will be _____________

To determine the vertical intercept:

The horizontal intercept (x-intercept) is the point at which the graph crosses the horizontal axis.

The output value of the horizontal intercept is always_________

The coordinates of the horizontal intercept will be ___________

To determine the horizontal intercept:

Example 1: Determine the vertical and horizontal intercepts for y = 3x – 2.

Example 2: Determine the vertical and horizontal intercepts for 4x – 2y = 10.


Page 160

You Try

5. Complete the table below. Write the intercepts as ordered pairs.

Equation Vertical Intercept Horizontal Intercept

y = 24 – 6x

5x – 3y = 30


Page 161

Section 8.6: Horizontal and Vertical Lines

Horizontal Lines y = b, where b is a real constant

Example 1: Graph the equation y = 2

x y Ordered Pair

Vertical Lines x = k, where k is a real constant

Example 2: Graph the equation x = –3

x y Ordered Pair

You Try

6.

a. Graph the equation y = –2

b. Graph the equation x = 4


Page 162

lesson 8: graphs and graphing linear equations...2014/06/09 · lesson 8: graphs and graphing...

Documents