Introduction to the Rectangular Coordinate System and Graphing Linear Equations

Plotting Points in the Rectangular Coordinate System

Understanding the Coordinate Axes

The rectangular coordinate system, also known as the Cartesian plane, is used to plot points, lines, and curves. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin (0,0).

• Horizontal axis (x-axis): Runs left to right.

• Vertical axis (y-axis): Runs up and down.

• Ordered pairs (x, y): Used to locate points on the plane.

The plane is divided into four quadrants:

• Quadrant I: (+, +)

• Quadrant II: (−, +)

• Quadrant III: (−, −)

• Quadrant IV: (+, −)

Graphing an Equation Using the Point-Plotting Method

Point-Plotting Method

To graph a linear equation, select integer values for x, compute the corresponding y values, and plot the resulting points. Connect the points with a straight line if the equation is linear.

• Step 1: Choose values for x (often from -3 to 3).

• Step 2: Substitute each x into the equation to find y.

• Step 3: Plot the points (x, y) on the coordinate plane.

• Step 4: Connect the points with a straight line.

Example: Graph $y = 4 - x$ for $x = -3$ to $x = 3$.

• For $x = -3$, $y = 4 - (-3) = 7$

• For $x = 0$, $y = 4 - 0 = 4$

• For $x = 3$, $y = 4 - 3 = 1$

Plot these points and draw a line through them.

Understanding the Viewing Rectangle

Setting Graphing Window Parameters

The viewing rectangle defines the visible portion of the coordinate plane on a graphing device or paper. It is determined by the minimum and maximum values for both axes and the scale (distance between tick marks).

Parameter	Description
Xmin	Minimum x-value (e.g., -10)
Xmax	Maximum x-value (e.g., 10)
Xscale	Distance between x-axis tick marks (e.g., 1)
Ymin	Minimum y-value (e.g., -10)
Ymax	Maximum y-value (e.g., 10)
Yscale	Distance between y-axis tick marks (e.g., 1)

Example: For a window of $[-10, 10, 1, -10, 10, 1]$, the x-axis and y-axis both range from -10 to 10 with tick marks every 1 unit.

Identifying Intercepts

X-Intercepts and Y-Intercepts

Intercepts are points where a graph crosses the axes:

• 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: For $y = 2x + 4$:

• Set $y = 0$ to find the x-intercept: $0 = 2x + 4 \implies x = -2$

• Set $x = 0$ to find the y-intercept: $y = 2(0) + 4 = 4$

Intercepts help in quickly sketching the graph of a linear equation.

Interpreting Information Given by Graphs

Reading and Analyzing Graphs

Graphs, especially line graphs, are used to represent relationships between variables. The horizontal axis (x-axis) often represents time or an independent variable, while the vertical axis (y-axis) represents the dependent variable.

• Trends: Upward or downward movement shows increase or decrease.

• Comparisons: Multiple lines can compare different data sets.

• Interpretation: Use the graph to answer questions about values, rates of change, and relationships.

Example: A graph showing the percentage of marriages ending in divorce by age at marriage. The graph can be used to estimate the divorce rate for a given age and to compare trends between different age groups.

Homework Practice: Solving and Graphing Linear Equations

Practice Problems

Solving for y given x in linear equations, and vice versa, is essential for graphing and understanding relationships between variables.

• Given $y = 3x + 4$, find $y$ for $x = -3$: $y = 3(-3) + 4 = -9 + 4 = -5$

• Given $y = -2x + 1$, find $y$ for $x = 2$: $y = -2(2) + 1 = -4 + 1 = -3$

• Given $y = x - 1$, find $x$ when $y = 4$: $4 = x - 1 \implies x = 5$

Practice with these types of problems builds fluency in graphing and interpreting linear equations.