College Algebra Study Guide: Equations, Functions, and Applications

Equations and Their Solutions

Solving Polynomial and Rational Equations

Equations are mathematical statements that assert the equality of two expressions. In College Algebra, students learn to solve various types of equations, including polynomial, rational, and radical equations.

• Polynomial Equations: Equations involving terms with variables raised to whole number powers. Example: $y = x^3 - 2x^2 + 3$

• Rational Equations: Equations containing fractions with polynomials in the numerator and/or denominator. Example: $\frac{2x}{x^2 - 4} = \frac{4}{x^2 - 4} - \frac{3}{x^2}$

• Radical Equations: Equations involving roots, such as square roots. Example: $\sqrt{x - 3} = 0$

• Absolute Value Equations: Equations involving the absolute value function. Example: $|2x - 4| + 2 = 8$

Key Steps for Solving:

1. Isolate the variable on one side of the equation.

2. Simplify expressions and combine like terms.

3. For quadratic equations, use factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

4. Check for extraneous solutions, especially with rational and radical equations.

Example: Solve $4(x + 2) - 3(5 - 2x) = 6 + 2x$ algebraically.

Functions and Their Properties

Function Notation and Evaluation

A function is a relation that assigns each input exactly one output. Functions are often written as $f(x)$, where $x$ is the input variable.

• Evaluating Functions: Substitute the given value into the function. Example: If $f(x) = x^2 - 2x$, then $f(2) = 2^2 - 2 \times 2 = 0$.

• Transformations: Functions can be shifted, stretched, or reflected. For example, $f(x - 4)$ shifts the graph right by 4 units.

Example: Find $f(2x)$ and $f(x - 4)$ for a given function $f(x)$.

Domain of Functions

The domain of a function is the set of all possible input values (typically $x$) for which the function is defined.

• For rational functions, exclude values that make the denominator zero.

• For radical functions, ensure the expression under the root is non-negative (for even roots).

Example: Find the domain of $f(x) = \frac{3}{x^2 + 4x - 12}$.

Linear Equations and Applications

Writing Equations of Lines

Lines in the plane can be described by equations in various forms, such as slope-intercept ($y = mx + b$) and point-slope ($y - y_1 = m(x - x_1)$).

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Have slopes that are negative reciprocals.

• Finding the Equation: Use given points or conditions to determine the slope and intercepts.

Example: Find the equation of a line parallel to $3x + 4y = 2$ through the point $(4, 1)$.

Linear Models in Applications

Linear equations are used to model real-world situations, such as pricing, speed, and mixture problems.

• Mixture Problems: Combine solutions of different concentrations to achieve a desired mixture.

• Distance, Rate, and Time: Use $\text{Distance} = \text{Rate} \times \text{Time}$ to solve travel problems.

• Cost Models: Example: $C = 12.76 + 0.18x$ models monthly electric charges.

Example: An electric company charges a flat fee plus a rate per kilowatt-hour. Write the equation for monthly cost.

Systems of Equations

Solving Systems of Linear Equations

A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Methods: Substitution, elimination, and graphical methods.

• Applications: Used to solve mixture, investment, and motion problems.

Example: Solve the system:

• $2x - y = 5$

• $3x - 3y = 9$

Quadratic Functions and Applications

Quadratic Equations and Their Graphs

Quadratic equations are of the form $y = ax^2 + bx + c$. Their graphs are parabolas.

• Intercepts: Points where the graph crosses the axes.

• Vertex: The highest or lowest point of the parabola.

• Applications: Used to model projectile motion, such as the height of a rock thrown upward.

Example: The height of a rock after $t$ seconds: $y = 20 - 4.9t^2$.

Variation and Proportionality

Direct and Inverse Variation

Variation describes how one quantity changes in relation to another.

• Direct Variation: $y = kx$ (as $x$ increases, $y$ increases proportionally)

• Inverse Variation: $y = \frac{k}{x}$ (as $x$ increases, $y$ decreases)

Example: The volume of a gas varies inversely with its pressure: $V = \frac{k}{P}$.

Function Operations and Composition

Operations on Functions

Functions can be added, subtracted, multiplied, divided, and composed.

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(fg)(x) = f(x) \cdot g(x)$

• Composition: $(f \circ g)(x) = f(g(x))$

Example: If $f(x) = 8x - 5$ and $g(x) = x^2 - 2x$, find $(fg)(x)$ and $(g - f)(x)$.

Mixture and Solution Problems

Solving Mixture Problems

Mixture problems involve combining solutions of different concentrations to achieve a desired concentration.

• Set up equations based on the amount of pure substance in each solution.

• Solve for unknown quantities using systems of equations.

Example: How much water should be added to 300 mL of a 60% acid solution to make a 50% solution?

Application Problems

Distance, Rate, and Time

Problems involving motion use the relationship $\text{Distance} = \text{Rate} \times \text{Time}$.

• Relative Speed: When moving against or with a current, adjust the rate accordingly.

• Multiple Travelers: Set up equations for each traveler and solve for unknowns.

Example: A motorboat travels upstream and downstream with different times due to the current. Find the speed of the boat.

Summary Table: Types of Equations and Methods

Type of Equation	Example	Method of Solution
Linear	$3x + 4y = 2$	Isolate variable, graph, substitution/elimination
Quadratic	$x^2 - 2x + \frac{3}{2}x + 4 = 0$	Factoring, quadratic formula, graphing
Rational	$\frac{2x}{x^2 - 4} = \frac{4}{x^2 - 4} - \frac{3}{x^2}$	Find common denominator, solve, check for extraneous solutions
Radical	$\sqrt{x - 3} = 0$	Isolate radical, square both sides, solve
Absolute Value	$|2x - 4| + 2 = 8$	Isolate absolute value, set up two cases, solve

Additional info:

• Some problems involve graphical solutions using calculators or graphing utilities.

• All equations should be solved to two decimal places when specified.

• Mixture and variation problems are common applications in College Algebra.