Graphs and Graphing Utilities in Precalculus

Chapter 1: Functions and Graphs

Section 1.1: Graphs and Graphing Utilities

This section introduces the foundational concepts of graphing in the rectangular coordinate system, including plotting points, graphing equations, interpreting graphing utility windows, and identifying intercepts. These skills are essential for understanding functions and their graphical representations in precalculus.

Rectangular Coordinate System

Definition and Structure

The rectangular coordinate system (also known as the Cartesian plane) is formed by two perpendicular number lines:

• x-axis: The horizontal axis.

• y-axis: The vertical axis.

• The point where the axes intersect is called the origin, denoted as (0, 0).

Points on the plane are identified by ordered pairs (x, y), where:

• x-coordinate: Distance and direction from the origin along the x-axis.

• y-coordinate: Distance and direction from the origin along the y-axis.

Positive and Negative Coordinates

• Positive x-values: To the right of the origin.

• Positive y-values: Above the origin.

• Negative x-values: To the left of the origin.

• Negative y-values: Below the origin.

Plotting Points in the Rectangular Coordinate System

Procedure for Plotting Points

To plot a point (x, y):

• Start at the origin (0, 0).

• Move x units left/right (negative/positive x).

• Move y units up/down (positive/negative y).

Example 1a: Plotting (-2, 4)

• Move 2 units to the left of the origin (x = -2).

• Move 4 units up (y = 4).

• Mark the point at (-2, 4).

Example 1b: Plotting (4, -2)

• Move 4 units to the right of the origin (x = 4).

• Move 2 units down (y = -2).

• Mark the point at (4, -2).

Graphs of Equations

Equations in Two Variables

A graph of an equation in two variables (x and y) represents all ordered pairs (x, y) that satisfy the equation. For example:

• Equation: $y = 4 - x^2$

• Each solution (x, y) makes the equation true when substituted.

Point-Plotting Method

To graph an equation using the point-plotting method:

1. Select values for x.

2. Calculate the corresponding y values.

3. Plot the resulting points (x, y) on the coordinate plane.

4. Connect the points to reveal the graph.

Example 3: Graphing $y = |x + 1|$

• Choose integer values for x (e.g., -4 to 2).

• Calculate y for each x:

x	y = |x + 1|	Ordered Pair (x, y)
-4	3	(-4, 3)
-3	2	(-3, 2)
-2	1	(-2, 1)
-1	0	(-1, 0)
0	1	(0, 1)
1	2	(1, 2)
2	3	(2, 3)

Plot these points and connect them to form the graph.

Graphing Utilities

Definition and Use

Graphing utilities include graphing calculators and computer software that allow users to graph equations and analyze their properties. The viewing rectangle sets the minimum and maximum values for both axes, determining the portion of the graph displayed.

• Standard viewing rectangle: $[-10, 10]$ for both x and y axes.

• Custom viewing rectangles can be set for specific needs.

Example: Understanding the Viewing Rectangle

Parameter	Value
Xmin	-100
Xmax	100
Xscale	50 (distance between tick marks)
Ymin	-100
Ymax	100
Yscale	10 (distance between tick marks)

Intercepts

Definitions

• x-intercept: The x-coordinate where the graph crosses the x-axis ($y = 0$).

• y-intercept: The y-coordinate where the graph crosses the y-axis ($x = 0$).

Example 5: Identifying Intercepts

• If the graph crosses the x-axis at (3, 0), the x-intercept is 3.

• If the graph crosses the y-axis at (0, 5), the y-intercept is 5.

Interpreting Information from Graphs

Application Example

Graphs can be used to interpret real-world data. For instance, the equation $d = 1.8n + 14$ models the percentage $d$ of marriages ending in divorce after $n$ years for high school graduates with no college education.

• To find the percentage after 15 years:

$d = 1.8(15) + 14 = 27 + 14 = 41$ So, 41% of marriages end in divorce after 15 years for this group.

Summary Table: Key Concepts

Concept	Definition	Example
Rectangular Coordinate System	Two perpendicular axes (x and y) intersecting at the origin	(-2, 4)
Ordered Pair	(x, y) specifies a point's location	(4, -2)
Equation in Two Variables	Relates x and y	$y = 4 - x^2$
Graphing Utility	Tool for graphing equations	Calculator, software
x-intercept	Where graph crosses x-axis ($y = 0$)	(3, 0)
y-intercept	Where graph crosses y-axis ($x = 0$)	(0, 5)

Additional info: These notes expand on the brief points in the slides to provide full academic context, definitions, and examples suitable for Precalculus students.