Section 1.1: Graphs and Graphing Utilities

Key Objectives

• Plotting points in the rectangular coordinate system: Understand how to accurately place points on a graph.

• Graphing equations in the rectangular coordinate system: Learn how to represent mathematical relationships visually.

• Using graphs to determine intercepts: Identify where graphs intersect the axes, providing crucial information about the functions.

• Interpreting information provided by graphs: Analyze graphs to extract useful information about the relationships they depict.

Rectangular Coordinate System

The rectangular coordinate system consists of two perpendicular axes:

• X-axis: The horizontal line that represents the independent variable in a relation.

• Y-axis: The vertical line that represents the dependent variable in a relation.

• Origin: The intersection point of the x and y axes, typically designated as (0, 0), which serves as the reference point for plotting other points.

Plotting Points

Each point in the coordinate system corresponds to a unique ordered pair (a, b):

• First coordinate (x-coordinate): Indicates the position along the x-axis.

• Second coordinate (y-coordinate): Indicates the position along the y-axis.

Example of Plotting a Point

To plot the point (-2, 4):

1. Move 2 units left from the origin to -2 on the x-axis.

2. From this position, move 4 units up to reach the y-coordinate of 4.The point is plotted at the intersection of these movements.

Graphs of Equations

A relationship between two quantities can be represented as an equation in two variables, such as y = mx + b.The solution set consists of all ordered pairs (x, y) that satisfy the equation, forming a continuous line or curve. The graph visually represents this solution set in the rectangular plane, illustrating how changes in one variable affect the other.

Examples of Graphing Equations

• Linear Equations: Graphing a linear equation results in a straight line.

• Quadratic Equations: Graphing quadratic equations produces a parabolic curve (U-shaped).

Exercises

1. Exercise 1: Graph the equation y - 5x = 2. Consider the points (-2, 21), (10, 2), and (3, X). Calculate the missing value for X when x = 3.

2. Exercise 2: Graph the equation x² + y = 1, depicting a curve.

3. Exercise 3: Graph y = |x + 1|, which forms a V-shaped graph reflecting the absolute value function.

Intercepts

• X-intercept: The x-coordinate of a point where the graph intersects the x-axis. Correspondingly, the y-coordinate at this point is always zero. To find the x-intercept, set y = 0 and solve the equation.

  • Example: For the equation y = 2x - 4, setting y = 0 gives x = 2, resulting in the x-intercept (2, 0).

• Y-intercept: The y-coordinate of a point where the graph intersects the y-axis. Here, the x-coordinate is always zero. To find the y-intercept, set x = 0 and solve the equation.

  • Example: For the equation y = 3x + 1, setting x = 0 gives y = 1, providing the y-intercept (0, 1).

Additional Exercise

• Identify both the x-intercept and y-intercept for a given equation:

  • X-intercept: (X, 0)

  • Y-intercept: (0, Y)Exercise: Given the equation y = -x + 5, determine both intercepts.

Understanding graphs and their properties is essential for visualizing relationships in mathematics and for solving real-world problems effectively.