Pre-Calculus – Key Concepts & Review Notes

What is Pre-Calculus?
Pre-Calculus is basically the class that connects algebra to calculus. It focuses a lot on understanding functions, how they look on graphs, and how different equations behave. The goal isn’t just memorizing formulas, it’s actually understanding what’s going on.

Functions
A function is a relationship where every input (x-value) has only one output (y-value).

Functions are usually written like:
f(x) = something

Common types of functions in pre-calculus include linear, quadratic, polynomial, rational, exponential, and logarithmic functions.

Graphs of Functions
Graphs show how a function behaves.

Things you usually need to find on a graph:

  • x-intercepts

  • y-intercept

  • domain (all possible x-values)

  • range (all possible y-values)

  • where the graph increases or decreases

  • maximums and minimums

Understanding graphs is a big part of pre-calculus.

Linear Functions
Linear functions make straight lines.

Standard form:
y = mx + b

m = slope (rate of change)
b = y-intercept

The slope tells you how steep the line is and whether it goes up or down.

Quadratic Functions
Quadratic functions create parabolas.

Standard form:
y = ax² + bx + c

Important things to know:

  • If a is positive, the parabola opens up

  • If a is negative, it opens down

  • The vertex is the highest or lowest point

  • The axis of symmetry goes through the vertex

Polynomial Functions
Polynomial functions can have more than one term and different powers.

Example:
f(x) = x³ − 2x² + x − 5

Important ideas:

  • Degree of the polynomial

  • End behavior

  • Zeros (also called roots)

Rational Functions
Rational functions are fractions with polynomials.

Example:
f(x) = (x + 1) / (x − 2)

Things to watch for:

  • Values that make the denominator zero are not allowed

  • Vertical asymptotes

  • Horizontal asymptotes

  • Holes in the graph

Exponential Functions
Exponential functions involve growth or decay.

General form:
y = a(b)^x

If b is greater than 1, it’s growth.
If b is between 0 and 1, it’s decay.

These are used for things like population growth and money problems.

Why Pre-Calculus Matters
Pre-Calculus helps build the foundation for calculus by teaching how functions work and how to analyze graphs. If you understand these concepts, calculus is way easier later on.