Graphs and Functions

The Rectangular Coordinate System (Cartesian Plane)

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. This system is fundamental for graphing equations and visualizing relationships between variables in algebra.

• Axes: The x-axis is horizontal, and the y-axis is vertical. They intersect at the origin (0, 0).

• Ordered Pairs: Points are represented as (x, y), where x is the horizontal position and y is the vertical position.

• Quadrants: The axes divide the plane into four quadrants, numbered counterclockwise starting from the upper right.

Example: The graph below shows the four quadrants of the Cartesian plane.

Example: Plot the points A (4, 3), B (–3, 2), C (–2, –3), D (5, –4), E (0, 0), F (0, –3) on the graph and identify the quadrant of each point.

Practice: Plot the points W (1, –2), X (5, 2), Y (–3, –4), Z (–4, 3) on the graph below.

Determining If an Ordered Pair Is a Solution

An ordered pair (x, y) is a solution to a two-variable equation if, when x and y are substituted into the equation, the equation is true.

• One-variable equation: x + 2 = 5. Solution: x = 3.

• Two-variable equation: x + 2y = 5. Solution: (x, y) pairs that satisfy the equation.

Example: For the equation x + 2y = 5, check if (5, 0), (8, 2), and (–3, 4) are solutions by substituting the values into the equation.

Completing Ordered Pair Solutions

To complete an ordered pair solution for a two-variable equation, substitute the given value for one variable and solve for the other.

• Example: For y = 2x + 3, if x = 2, then y = 2(2) + 3 = 7, so (2, 7) is a solution.

• Example: For x = 3y – 2, if y = 5, then x = 3(5) – 2 = 13, so (13, 5) is a solution.

Graphing Linear Equations in Two Variables

To graph a linear equation in two variables (Ax + By = C):

1. Find ordered pair solutions by plugging in x-values to find y-values.

2. Plot those ordered pairs and connect them with a straight line.

Example: Graph the equation y = 4x – 3 by plotting points and connecting them on the graph.

Intro to Relations and Functions

A relation is a set of ordered pairs (x, y). A function is a relation where each input (x) is paired with at most one output (y).

• Relations can be represented as sets of ordered pairs, tables, graphs, or equations.

• Functions are special relations with unique outputs for each input.

Vertical Line Test

The vertical line test is used to determine if a graph represents a function. If any vertical line crosses the graph more than once, the relation is not a function.

Domain and Range

The domain of a relation or function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Domain and range can be listed as sets or written in interval notation.

Function Notation

If an equation is a function, it can be written as y = f(x), read as "f of x." This notation emphasizes the dependence of y on x.

• Example: If f(x) = 3x – 1, then f(4) = 3(4) – 1 = 11.

Graphing Linear Equations Using Intercepts

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

• To find the x-intercept, set y = 0 and solve for x.

• To find the y-intercept, set x = 0 and solve for y.

Slope of a Line

The slope (m) of a line measures its steepness and is calculated as the change in y divided by the change in x between two points:

• Positive slope: Line rises from left to right.

• Negative slope: Line falls from left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Slope-Intercept Form

The slope-intercept form of a line is , where m is the slope and b is the y-intercept.

• To graph, plot the y-intercept (0, b) and use the slope to find another point.

Parallel and Perpendicular Lines

• Parallel lines: Have the same slope but different y-intercepts.

• Perpendicular lines: Slopes are negative reciprocals of each other (i.e., ).

Point-Slope Form

The point-slope form of a line is , where (x₁, y₁) is a point on the line and m is the slope.

• Use this form when given a point and the slope.

Linear Inequalities in Two Variables

A linear inequality in two variables divides the plane into a region of solutions. The boundary is a line, which is solid for ≤ or ≥ and dashed for < or >.

• To graph, first draw the boundary line, then shade the region that satisfies the inequality.

• Test a point (often the origin) to determine which side to shade.

Example: Graph the inequality by drawing the line (dashed) and shading above the line.