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Section 7.1, Slide *

CHAPTER 7

Algebra: Graphs, Functions, and Linear Systems

Section 7.1, Slide *

7.1

Graphing and Functions

Section 7.1, Slide *

Objectives

Plot points in the rectangular coordinate system.
Graph equations in the rectangular coordinate system.
Use function notation.
Graph functions.
Use the vertical line test.
Obtain information about a function from its graph.

Section 7.1, Slide *

Points and Ordered Pairs

The horizontal number line is the x-axis.

The vertical number line is the y-axis.

The point of intersection of these axes is their zero point, called the origin.

Negative numbers are shown to the

left of and below the origin.

The axes divide the plane into four

quarters called “quadrants”.

Section 7.1, Slide *

Points and Ordered Pairs

Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers, (x, y).

Look at the ordered pairs

(−5, 3) and (3, −5).

The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis.

The second number in each pair, called the y-coordinate, denotes the vertical distance and direction along the x-axis or parallel to it.

The figure shows how we plot, or locate the points corresponding to the ordered pairs.

Section 7.1, Slide *

Example: Plotting Points in the Rectangular Coordinate System

Plot the points: A(−3, 5), B(2, −4), C(5,0), D(−5,−3), E(0, 4), and F(0, 0).

Solution: We move from the origin and plot the point in the following way:

A(-3,5):	3 units left, 5 units up
B(2,4):	2 units right, 4 units down
C(5,0):	5 units right, 0 units up or down
D(-5,-3):	5 units left, 3 units down
E(0,4):	0 units left or right, 4 units up
F(0,0):	0 units left or right, 0 units up or down

Section 7.1, Slide *

Graphs of Equations

A relationship between two quantities can be expressed as an equation in two variables, such as

y = 4 – x2.

A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property:

When the x-coordinate is substituted for x and the y coordinate is substituted for y in the equation, we obtain a true statement.

The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation.

Section 7.1, Slide *

Example: Graphing an Equation Using the Point-Plotting Method

Graph y = 4 – x2. Select integers for x, starting with −3 and ending with 3.

Solution: For each value of x, we find the corresponding value for y.

Section 7.1, Slide *

Example continued

Now plot the seven points and join them with a smooth curve.

Section 7.1, Slide *

Functions

If an equation in two variables (x and y) yields precisely one value of y for each value of x, we say that y is a function of x.

The notation y = f(x) indicates that the variable y is a function of x. The notation f(x) is read “f of x.”

Section 7.1, Slide *

Example: Graphing Functions

Graph the functions f(x) = 2x and g(x) = 2x + 4 in the same rectangular coordinate system. Select integers for x from −2 to 2, inclusive.

Solution: For each function we use tables to display the coordinates:

Section 7.1, Slide *

Example continued

Next, plot the five points for each function and connect them.

Do you see a relationship between the two graphs?

Section 7.1, Slide *

Vertical Line Test

Section 7.1, Slide *

Example: Using the Vertical Line Test

Use the vertical line test to identify graphs in which y is a function of x.

a. b. c. d.

Section 7.1, Slide *

Example continued

Solution: y is a function of x for the graphs in (b) and (c).

a. b. c. d.

Intersects the graph twice, so y is not a function.

Intersects the graph once, so the graph defines a function.

Section 7.1, Slide *

Example: Analyzing the Graph of a Function

The given graph illustrates the body temperature from 8 a.m. through 3 p.m. Let x be the number of hours after 8 a.m. and y be the body temperature at time x.

What is the temperature at 8 a.m.?
During which period of time is your temperature decreasing?
Estimate your minimum temperature during the time period shown. How many hours after 8 a.m. does this occur? At what time does this occur?

Section 7.1, Slide *

Example continued

d. During which period of time is your

temperature increasing?

Part of the graph is shown as a horizontal line segment. What does this mean about your temperature and when does this occur?
Explain why the graph defines y as a function of x.

Section 7.1, Slide *

Example continued

Solution:

a. The temperature at 8 a.m. is when x is 0, since no time has passed when it is 8 a.m. Thus, the temperature at 8 a.m. is 101°F.

The temperature is decreasing

when the graph falls from left to

right. This occurs between x = 0

and x = 3. Thus, the temperature

is decreasing between the times

8 a.m. and 11 a.m.

Section 7.1, Slide *

Example continued

c. The minimum temperature can be found by locating the lowest point on the graph. This point lies above 3 on the horizontal axis. The y-coordinate falls

midway between 98 and 99, at ap-

proximately 98.6. Thus, the minimum

temperature is 98.6°F at 11 a.m.

The temperature is increasing when

the graph rises from left to right.

This occurs between x = 3 and x = 5. Thus, the temperature is increasing from 11 a.m. to 1 p.m.

Section 7.1, Slide *

Example continued

e. The horizontal line segment shown

indicates that the temperature is

neither increasing nor decreasing.

The temperature remains the same

at 100°F, between x = 5 and x = 7.

Thus, the temperature is at a constant

100°F between 1 p.m. and 3 p.m.

f. By the vertical line test we can see that no vertical line will intersect the graph more than once. So, the body temperature is a function of time. Each hour represents one body temperature.