Graphs of Equations

Rectangular Coordinate System (Cartesian Plane)

The rectangular coordinate system, also known as the Cartesian Plane, is a fundamental tool for graphing equations in two dimensions. It consists of two perpendicular axes that intersect at the origin, dividing the plane into four quadrants.

• Axes: The horizontal axis is the x-axis, and the vertical axis is the y-axis.

• Ordered Pairs: Points are represented as ordered pairs (x, y).

• Origin: The point (0, 0) where the x and y axes intersect.

• Quadrants: The axes divide the plane into four quadrants, numbered counter-clockwise starting from the top-right.

• Signs: x-values are positive to the right of the origin and negative to the left; y-values are positive above the origin and negative below.

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

Equations of Two Variables

Many equations in algebra involve two variables, typically x and y. The graph of such an equation is the set of all points (x, y) that satisfy the equation.

• Checking Solutions: Substitute x and y values into the equation to determine if a point satisfies it.

• Graph Representation: Points that satisfy the equation are on its graph; those that do not are not on the graph.

Example: For the equation , check if (3, 2), (4, 1), (0, 0), and (−1, 3) satisfy the equation.

Graphing Two Variable Equations by Plotting Points

To graph an equation with two variables, calculate and plot points that make the equation true. This method is called graphing by plotting points.

1. Isolate y: Rewrite the equation so y is on the left side.

2. Choose 3–5 x-values and calculate corresponding y-values.

3. Plot the (x, y) points.

4. Connect the points with a line or curve.

Example: Graph by choosing x = −2, −1, 0, 1, 2 and calculating y for each.

Example: Graph by isolating y and plotting points.

Example: Graph by choosing positive x-values.

Graphing Intercepts

Intercepts are points where a graph crosses the axes. They are useful for quickly sketching graphs.

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

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

• Ordered Pairs: Intercepts are written as ordered pairs.

Example: For a given graph, identify the x-intercept and y-intercept.

Lines: Slope and Types of Slope

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y to the change in x between two points.

• Slope Formula:

• Types of Slope:

  • Positive: Line goes up from left to right.

  • Negative: Line goes down from left to right.

  • Zero: Horizontal line ().

  • Undefined: Vertical line ().

Example: Find the slope of the line containing (1, 1) and (4, 3).

Slope-Intercept Form

The slope-intercept form of a line is a convenient way to write and graph linear equations.

• General Form:

• m: Slope of the line

• b: y-intercept (the value of y when x = 0)

Example: Identify the slope and y-intercept of and graph the line.

Graphing Lines from Slope-Intercept Form

To graph a line given in slope-intercept form:

1. Plot the y-intercept (0, b).

2. Use the slope to find a second point: rise/run from the y-intercept.

3. Draw a line through the points.

Example: Graph by plotting (0, 3) and using the slope m = 2.

Point-Slope Form

The point-slope form is used when you know the slope and a point on the line (not necessarily the y-intercept).

• General Form:

• Useful for writing equations given a point and slope or two points.

Example: Write the equation of a line with slope 5 passing through (1, 3):

Finding Equations of a Line Given Two Points

To find the equation of a line given two points:

1. Calculate the slope using the two points.

2. Use point-slope form with one of the points.

3. Rewrite in slope-intercept form if needed.

Example: Find the equation of the line passing through (−1, 5) and (2, 4).

Standard Form of a Line

The standard form of a line is . To find slope or intercepts, rewrite the equation or set variables to zero.

• To find y-intercept: Set x = 0 and solve for y.

• To find x-intercept: Set y = 0 and solve for x.

• To find slope: Rewrite in slope-intercept form.

Example: For , set x = 0 for y-intercept and y = 0 for x-intercept.

Parallel and Perpendicular Lines

Parallel and perpendicular lines are distinguished by their slopes.

Example: Write the equation of a line perpendicular to with y-intercept 3.

Example: Write the equation of a line passing through (−1, 4) that is parallel to .

Summary Table: Forms of Linear Equations

Additional info: The notes include practice problems and examples for each concept, reinforcing the methods for graphing and analyzing lines and equations in the coordinate plane.