Precalculus Cheat Sheet

This cheat sheet covers foundational concepts in Precalculus, including sets, relations, functions, graphing, and transformations, providing a concise overview of key definitions, theorems, and examples.

Core Principles

A set is a well-defined collection of objects called elements.
A relation is a set of points in the plane.
A function is a relation where each x-coordinate is matched with only one y-coordinate.
The Vertical Line Test determines if a graph represents y as a function of x.
Transformations (shifts, reflections, scalings) alter a function's graph based on changes to its input or output.

Action Steps

To find x-intercepts, set y=0 in the equation and solve for x.
To find y-intercepts, set x=0 in the equation and solve for y.
To test for y-axis symmetry, substitute (-x, y) into the equation and simplify.
To test for origin symmetry, substitute (-x, -y) into the equation and simplify.
To graph a function, plot key points (intercepts, extrema) and connect them smoothly, considering transformations.

Formulas

Distance Formula: $d = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$
Midpoint Formula: $M = \left(\frac{x_0 + x_1}{2}, \frac{y_0 + y_1}{2}\right)$
Symmetry about x-axis: $(a, b)$ and $(c, d)$ are symmetric if $a=c$ and $b=-d$.
Symmetry about y-axis: $(a, b)$ and $(c, d)$ are symmetric if $a=-c$ and $b=d$.
Symmetry about origin: $(a, b)$ and $(c, d)$ are symmetric if $a=-c$ and $b=-d$.

Key Terms

Set: A well-defined collection of objects, called elements.
Relation: A set of points in the plane.
Function: A relation where each x-coordinate is matched with only one y-coordinate.
Domain: The set of all possible x-coordinates of the points in a function.
Range: The set of all possible y-coordinates of the points in a function.
Intercepts: Points where a graph crosses or touches the x-axis (x-intercepts) or y-axis (y-intercepts).
Symmetry: Properties of a graph that remain unchanged when reflected across the x-axis, y-axis, or origin.
Transformations: Operations (shifts, reflections, scalings) that change a function's graph without altering its fundamental shape.

Real World Examples

Cost of grapes based on weight: The cost C(g) of g pounds of grapes at $1.50 per pound is modeled by C(g) = 1.5g.
Height of a model rocket: The height h(t) in feet of a rocket t seconds after lift-off is modeled by a piecewise function: h(t) = -5t^2 + 100t for 0 ≤ t ≤ 20, and h(t) = 0 for t > 20.
Economic functions (price-demand, revenue, cost, profit): These functions model relationships between production levels, prices, costs, and profits in business scenarios.