Unit 1 Precalculus

Key Vocabulary Terms and Questions for Unit 1 AP Precalculus Test

Relation

A set of ordered pairs, where each pair consists of an input and an output value, often representing a relationship between two quantities.

Function

A specific type of relation where each input is associated with exactly one output in the range, typically described by a formula or equation.

Polynomial Function

The graph crosses the x-axis up to n times and has up to n-1 vertices. The domain is all real numbers. The range would be the values greater/lesser than the minimum/maximum points.

Quadratic Function

Form: y = ax²+bx+c

The graph changes direction at its one vertex. The domain is all real numbers. The range would be the values greater/lesser than the vertex depending on whether it is a minimum or maximum point.

Linear Function

Form: y = ax+b

The straight-line graph, f(x), changes at a constant rate as x changes. The domain is all real numbers. The range is not always real numbers because if the slope is 0, the range will only be b.

Differences Table

Way of analyzing a function by looking at its rate of change. One must have equal length input intervals to use a differences table. We don’t use slope, but something similar since we are essentially calculating rates of change.

Direct Variation

The straight-line graph goes through the origin. The domain is all real numbers. It can be ANY power of x. It can be a quadratic or cubic or anything that goes through the origin.

Power Function

Form: y = ax^b

The domain depends on the value of b. It is completely dependent on the b number on what the graph will look like. For positive integer values of b, the domain is all real numbers; for negative integer balues of b, the domain is x ≠ 0.

Inverse Function

Form: y = a/x or a/x^-n

f(x) varies inversely with x (or with the nth power of x). For positive values of n, domain is x ≠ 0. For real-world applications, domain is x > 0.

Rational Function

Form: y = n(x)/d(x) where n and d are polynomial functions

Has a discontinuity (asymptote or missing point) where the denominator is 0; it may have horizontal or other asymptotes.

Exponential Function

Form: y = a*b^x where a ≠ 0, b > 0, b ≠ 1

f(x) varies exponentially with x. Graph crosses the y-axis at f(0) = a and has the x-axis as an asymptote.

Differences Table for Exponential Functions

The differences are dependent on the b factor. For example, if b was 2, the differences would all be 2 times the previous difference.

Boolean Variable

Equals 1 if a given condition is true and 0 if that condition is false.

Graphing Boolean Restrictions

Follow the images’ method

Increasing Function

If input values increase, the output values always increase. Given a and b are in the interval, if a < b, then f(a) < f(b).

Decreasing Function

If input values increase, the output values always decrease. Given a and b are in the interval, if a < b, then f(a) > f(b).

Concave Up

Rate of change is increasing.

Concave Down

Rate of change is decreasing.

Concave Up

Sample Table of Values.

Concave Down

Sample Table of Values.

Composition of Functions

Something like f(g(x)).

Inverses Rule

The graphs of inverses are reflections over the line y = x.

Parametric Function

Simple way to plot the graph of the inverse of a function. Here, x and y are both expressed in terms of some third variable, usually t which is known as the PARAMETER. The variable t is often the independent variable in real-world applications because it is time or such variable.

Parametric Equation Form

X1T = T, Y1T = original function (replace variable with T), X2T = original function (replace variable with T), Y2T = T. Note: Put domain in Tmin and Tmax in Window in calculator. And when dealing with quadratic business, always find out vertex which will help in finding the range.