Linear Functions, Equations, and Inequalities

Solving for Missing Lengths in Right Triangles

To find the missing side of a right triangle, use the Pythagorean Theorem, which relates the lengths of the sides:

• Pythagorean Theorem:

• Given two sides, solve for the third by rearranging the equation.

• Example: If , , then .

Midpoint and Endpoint Formulas

The midpoint formula finds the point exactly halfway between two endpoints in the coordinate plane. If one endpoint and the midpoint are known, the other endpoint can be found by solving the midpoint equations for the unknown coordinates.

• Midpoint Formula:

• Finding an Endpoint: If and are known, solve for using and .

• Example: If and , then .

Distance Formula

The distance between two points in the coordinate plane is found using the distance formula, derived from the Pythagorean Theorem.

• Distance Formula:

• Example: For and ,

Evaluating and Graphing Linear Functions

Linear functions can be evaluated by substituting values for . Their graphs are straight lines, and the slope-intercept form is useful for graphing and analysis.

• Function Evaluation: Substitute into to find .

• Slope-Intercept Form: , where is the slope and is the y-intercept.

• Example: , .

Finding the Slope and Equation of a Line

The slope of a line through two points is calculated as the change in over the change in . The point-slope form is used to write the equation of a line given a point and the slope.

• Slope Formula:

• Point-Slope Form:

• Example: Through and , ; equation:

Parallel and Perpendicular Lines

Parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.

• Parallel:

• Perpendicular:

• Example: Line through parallel to has .

Systems and Applications

Mixture Problems

Mixture problems involve combining solutions of different concentrations to achieve a desired concentration. Set up equations based on the total amount and the amount of pure substance.

• General Setup:

• Example: How much pure alcohol should be added to 20 L of 40% alcohol to get 50% alcohol?

• Equation:

• Solve: liters

Analysis of Graphs of Functions

Domain and Range

The domain of a function is the set of all possible input values (), and the range is the set of all possible output values (). Graphs help visualize these sets.

• Example: For a given graph, ,

Determining if a Relation is a Function

A relation is a function if each input has exactly one output. This can be checked by examining ordered pairs or using the vertical line test on a graph.

• Example: The set is a function; is not (repeated input with different outputs).

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry:

• Even Function: for all in the domain (symmetric about the -axis).

• Odd Function: for all in the domain (symmetric about the origin).

• Example: is odd because .

Further Topics in Algebra

Function Operations and Composition

Functions can be added, subtracted, multiplied, divided, or composed. The composition means .

• Example: If , , then , and .

Solving Absolute Value Equations and Inequalities

Absolute value equations are solved by considering both the positive and negative cases.

• Example: leads to , so or .

• Solutions: or .

Translation of Graphs

Graph translation involves shifting the graph of a function horizontally or vertically.

• Example: is the graph of shifted right by 6 units and down by 3 units.

Word Problems and Applications

Many algebraic concepts are applied to real-world problems, such as cost modeling and mixture problems.

• Example: A company charges a flat fee of f(x) = 5x + 20$.

Summary Table: Slope-Intercept and Point-Slope Forms