Algebra: Equations and Functions

Algebra is the language of patterns and relationships. It turns real situations into symbols you can manipulate, analyze, and use to predict outcomes. At the center of algebra are two closely connected ideas: equations, which express that two quantities are equal, and functions, which describe how one quantity depends on another. Mastering these topics builds the foundation for advanced mathematics, science, economics, computing, and any field where decisions depend on quantitative reasoning.

What an Equation Really Says

An equation is a statement that two expressions represent the same value:

• 3x + 5 = 17
• 2(x - 4) = x + 8

The goal is often to solve for a variable, meaning you find the value(s) of the variable that make the equation true.

Solving Linear Equations

A linear equation has variables only to the first power and produces a straight-line graph. Examples include 2x - 7 = 9 or 3y = 2x + 6. The key principle is to keep the equation balanced by performing the same operation on both sides.

Common steps:

1. Simplify both sides (distribute, combine like terms).
2. Move variable terms to one side and constants to the other.
3. Isolate the variable using inverse operations.

For example, with 4x - 3 = 2x + 9, subtract 2x from both sides to get 2x - 3 = 9, then add 3 to get 2x = 12, and divide by 2 to get x = 6.

Manipulating Expressions: The Toolkit

Before solving, you frequently need to rewrite expressions in equivalent forms. This is where algebra becomes practical rather than procedural.

Key skills include:

• Distributive property: a(b + c) = ab + ac
• Combining like terms: 2x + 5x = 7x
• Factoring: reversing distribution, such as turning x^2 + 5x into x(x + 5)

These tools matter because they reveal structure. Factoring, for example, can expose zeros of a polynomial or simplify rational expressions.

Functions: Relationships You Can Graph

A function is a rule that assigns each input exactly one output. If you write f(x) = 2x + 3, then for every x you plug in, you get a single result.

Functions are powerful because they let you:

• model change,
• compare rates,
• predict outputs,
• interpret graphs meaningfully.

Function Notation and Meaning

Function notation helps you keep track of inputs and outputs:

• f(4) means “the output when the input is 4.”
• f(x + 1) means “the output when the input is x + 1,” which requires careful substitution.

Understanding function notation is essential in later topics like transformations, calculus, and modeling.

Domain and Range

• The domain is the set of allowable inputs.
• The range is the set of resulting outputs.

In many school problems, the domain is assumed to be all real numbers unless restrictions appear. For example, a rational expression like g(x) = 1/(x - 2) cannot accept x = 2 because division by zero is undefined. These restrictions affect the graph and the real-world meaning of a model.

Linear Functions: Constant Rate of Change

A linear function is commonly written in slope-intercept form:

y = mx + b

• m is the slope, the rate of change.
• b is the y-intercept, the value when x = 0.

Slope is not just a graph feature; it is a relationship. If a taxi fare is __MATH_INLINE_0__2 per mile, then the model C(miles) = 2(miles) + 3 is linear. The slope (2) tells you the cost increases by $2 for each additional mile.

Other useful forms include:

• Point-slope form: y - y1 = m(x - x1) when you know a point and slope.
• Standard form: Ax + By = C, often used in systems of equations.

Graphing Linear Functions

To graph a line, you can:

• plot the y-intercept and use the slope as “rise over run,” or
• use two points by evaluating the function at two x-values.

Interpreting a graph matters as much as drawing it. Steeper slopes mean faster change, while a larger intercept shifts the line upward.

Systems of Equations: When Two Relationships Meet

A system of equations involves two or more equations with the same variables. Solutions represent values that satisfy all equations at once. Graphically, the solution is where the graphs intersect.

For two linear equations, three outcomes are possible:

• One solution: lines intersect once.
• No solution: lines are parallel.
• Infinitely many solutions: lines coincide (same line).

Methods for Solving Systems

Common approaches include:

• Graphing: useful for visual understanding and approximate solutions.
• Substitution: best when one equation is already solved for a variable.
• Elimination: efficient when variables align and can be canceled.

In real-world terms, systems model constraints. For example, if a school club sells tickets and must meet both a revenue goal and a ticket-count limit, the system represents the conditions that must hold simultaneously.

Quadratic Functions and Equations: Curves and Turning Points

Quadratic expressions involve a variable squared, such as:

y = ax^2 + bx + c

Quadratic graphs are parabolas. The coefficient a determines whether the parabola opens up (a > 0) or down (a < 0) and how “wide” it appears.

Solving Quadratic Equations

A quadratic equation is often written as:

ax^2 + bx + c = 0

Common solution methods include:

• Factoring: fastest when it works cleanly.
• Completing the square: reveals vertex form and supports deeper understanding.
• Quadratic formula: works for all quadratics.

Factoring example idea: if you can rewrite x^2 - 5x + 6 as (x - 2)(x - 3), then the solutions come from setting each factor to zero, giving x = 2 and x = 3.

Interpreting Quadratic Graphs

Key features of a quadratic function include:

• Vertex: the highest or lowest point of the parabola.
• Axis of symmetry: a vertical line through the vertex.
• Intercepts: where the graph crosses the x-axis (real solutions) and y-axis.

These features connect algebra to meaning. In a projectile model, the vertex corresponds to maximum height, and x-intercepts can represent when an object hits the ground.

Polynomials and Their Graphs

A polynomial is an expression like:

p(x) = 2x^3 - x^2 + 4x - 7

Polynomials generalize linear and quadratic functions. Their graphs can have multiple turning points, and their behavior for large positive or negative x depends heavily on:

• the degree (highest power),
• the leading coefficient (coefficient of the highest power term).

Understanding polynomial graphs is less about memorizing shapes and more about reading structure:

• zeros (x-intercepts) relate to factors,
• multiplicity affects whether the graph crosses or just touches the x-axis,
• end behavior comes from the leading term.

Word Problems: Turning Context into Algebra

Word problems are where equations and functions prove their value. The challenge is translating a situation into mathematical form.

A practical approach:

1. Identify quantities and assign variables.
2. Write relationships as equations or functions.
3. Solve and interpret the result in context.
4. Check whether the solution makes sense (units, reasonableness, constraints).

For example, “fixed fee plus per-unit cost” scenarios naturally produce linear functions. “Area problems” often lead to quadratics. “Two conditions at once” points to systems of equations.

Why These Skills Matter Beyond Algebra

Solving equations teaches disciplined reasoning: you transform a statement while preserving truth. Functions teach you to see relationships, not just numbers. Graphs make those relationships visible, allowing you to interpret trends, compare models, and understand behavior at a glance.

Linear functions describe constant change, quadratics describe situations with acceleration or curvature, systems model competing constraints, and polynomials build a bridge to more complex behavior. Together, equations and functions form the core of algebra and the launching point for nearly everything that comes next in mathematics.