뒤로Graphs and Graphing Utilities in Precalculus
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Graphs and Graphing Utilities
Introduction to the Rectangular Coordinate System
The rectangular coordinate system (also called the Cartesian plane) is a fundamental tool in algebra and precalculus for visualizing relationships between variables. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).
Positive numbers are to the right of the origin (x-axis) and above the origin (y-axis).
Negative numbers are to the left of the origin (x-axis) and below the origin (y-axis).
Plotting Points in the Rectangular Coordinate System
Each point in the plane is represented by an ordered pair (x, y):
The x-coordinate indicates the horizontal position (right for positive, left for negative).
The y-coordinate indicates the vertical position (up for positive, down for negative).
Example: To plot (−2, 4), move 2 units left and 4 units up from the origin.
Example: To plot (4, −2), move 4 units right and 2 units down from the origin.
Graphs of Equations
A graph of an equation in two variables (x and y) is the set of all points (x, y) that satisfy the equation. For example, the equation can be graphed by finding ordered pairs that make the equation true.
To graph an equation, select values for x, compute the corresponding y values, and plot the resulting points.
Example Table:
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 visualize the graph of the equation.
Graphing Utilities and Viewing Rectangles
Graphing utilities (such as calculators or software) allow you to plot equations quickly. The viewing rectangle determines the portion of the coordinate plane displayed, defined by minimum and maximum x and y values, and the scale between tick marks.
The standard viewing rectangle is typically [−10, 10, 1] by [−10, 10, 1].
For example, [−100, 100, 50] by [−100, 100, 10] means x and y range from −100 to 100, with tick marks every 50 units (x) and 10 units (y).
Intercepts
Intercepts are points where a graph crosses the axes:
x-intercept: Where the graph crosses the x-axis (y = 0).
y-intercept: Where the graph crosses the y-axis (x = 0).
Example: The graph crosses the x-axis at (−3, 0) and the y-axis at (0, 5).
Interpreting Information from Graphs
Graphs can model real-world situations. For example, the equation models the percentage of marriages ending in divorce after n years if the wife is under 18 at marriage.
To find the percentage after 15 years:
The point (15, 65) should appear on the graph of the equation.
Application: This process demonstrates how to use equations and their graphs to answer practical questions by interpreting points on the graph.
