AP Precalculus Formulae: Comprehensive Guide

Unit 1: Polynomial and Rational Functions

In Unit 1 of AP Precalculus, we explore the fundamentals of polynomial and rational functions. Key topics include how polynomials change over intervals, how to model rates of change in linear and quadratic functions, and how to analyze rational functions with vertical asymptotes, holes, and end behavior. We also emphasize transformations, equivalent representations, and function model selection.

1.1 Change in Tandem

“Change in tandem” typically refers to how two variables in a function or relationship change simultaneously. If we consider a function \( y = f(x) \), a small change \(\Delta x\) in \(x\) may produce a corresponding change \(\Delta y\) in \(y\). This concept is fundamental to analyzing how polynomial or rational functions behave as we move from one input value to another.

1.2 Rates of Change

A rate of change measures how a quantity \(y\) changes with respect to another quantity \(x\). For a function \( f(x) \), the average rate of change over an interval \([a, b]\) is:

In polynomial contexts, the average rate of change between two points can show us if a function is increasing or decreasing, and how quickly.

1.3 Rates of Change in Linear and Quadratic Functions

Linear Functions: If \( f(x) = mx + b \), the slope \( m \) is the constant rate of change, meaning \(\Delta y / \Delta x = m\) for any two distinct points.

Quadratic Functions: If \( f(x) = ax^2 + bx + c \), the rate of change is not constant. The average rate of change between \( x_1 \) and \( x_2 \) depends on the interval. For instance, \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{a x_2^2 + b x_2 + c - (a x_1^2 + b x_1 + c)}{x_2 - x_1}. \] Simplified, it expresses how the parabola’s slope evolves over intervals.

1.4 Polynomial Functions and Rates of Change

Higher-degree polynomial functions (e.g., cubic, quartic) display more complex behaviors of rate of change. However, the concept is the same: we look at how \(f(x)\) changes between intervals. In AP Precalculus, we often examine how these average rates of change hint at broader features like turning points and intervals of increase or decrease.

1.5 Polynomial Functions and Complex Zeros

A polynomial function can be written in standard form as: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, \] where \(a_n \neq 0\). The Fundamental Theorem of Algebra states that every non-constant, single-variable polynomial with complex coefficients has exactly as many complex roots (counting multiplicities) as its degree \(n\). Complex zeros may come in conjugate pairs if the coefficients are real.

1.6 Polynomial Functions and End Behavior

“End behavior” refers to how a polynomial function behaves as \( x \to \infty \) or \( x \to -\infty \). For instance:

• If \(n\) is even and \(a_n > 0\), \( P(x) \to +\infty \) as \( x \to \pm\infty \).
• If \(n\) is even and \(a_n < 0\), \( P(x) \to -\infty \) as \( x \to \pm\infty \).
• If \(n\) is odd and \(a_n > 0\), \( P(x) \to -\infty \) as \( x \to -\infty \) and \( +\infty \) as \( x \to +\infty \).
• If \(n\) is odd and \(a_n < 0\), \( P(x) \to +\infty \) as \( x \to -\infty \) and \(-\infty\) as \( x \to +\infty \).

1.7 Rational Functions and End Behavior

A rational function is of the form \[ R(x) = \frac{P(x)}{Q(x)}, \] where \(P\) and \(Q\) are polynomials and \(Q \neq 0\). End behavior depends on the degrees of \(P\) and \(Q\):

• If \(\deg(P) < \deg(Q)\), \( R(x) \to 0 \) as \( x \to \pm\infty \).
• If \(\deg(P) = \deg(Q)\), \( R(x) \to \frac{a_n}{b_m} \) as \( x \to \pm\infty \), where \(a_n\) and \(b_m\) are the leading coefficients.
• If \(\deg(P) > \deg(Q)\), the behavior might resemble a polynomial (possibly with an oblique asymptote).

1.8 Rational Functions and Zeros

The zeros (or x-intercepts) of a rational function \(R(x)\) occur where \(P(x)=0\), provided that \(Q(x)\neq 0\) at those points. Hence, solving \(P(x)=0\) is key to finding zeros of \(R(x)\).

1.9 Rational Functions and Vertical Asymptotes

Vertical asymptotes occur for \( x = a \) if \(Q(a)=0\) (denominator is zero) and \(P(a) \neq 0\). Near vertical asymptotes, the function typically grows without bound (positive or negative infinity).

1.10 Rational Functions and Holes

A hole in the graph occurs at \( x = a \) if both \(P(a)=0\) and \(Q(a)=0\), but the common factor that caused this can be canceled out from numerator and denominator. If the factor remains after simplification, it’s an asymptote; if it cancels, the graph has a removable discontinuity (a hole).

1.11 Equivalent Representations of Polynomial and Rational Expressions

Factoring, expanding, and simplifying are common ways to find equivalent forms. For instance, two expressions that differ by a factor that’s zero except at a hole still represent the same function graphically (apart from the hole).

1.12 Transformations of Functions

Transformations include vertical and horizontal shifts, stretches, compressions, and reflections. If \( f(x) \) is a base function, then transformations might look like \[ y = a \cdot f(b(x - h)) + k, \] controlling vertical stretch (\(a\)), horizontal stretch (\(\frac{1}{b}\)), horizontal shift (\(h\)), and vertical shift (\(k\)).

1.13 Function Model Selection and Assumption Articulation

In applied contexts, choosing a polynomial or rational function model depends on observed behavior—like end behavior, presence of asymptotes, or turning points. Articulating assumptions means clarifying why a particular model (polynomial vs. rational) is chosen for a data set.

1.14 Function Model Construction and Application

Model construction often involves using key data points (intercepts, asymptotes, turning points) to determine constants in a polynomial or rational function. Application typically involves interpreting domain/range restrictions, predicting behavior, or extrapolating future values.

Unit 2: Exponential and Logarithmic Functions

Unit 2 moves beyond polynomials and rational functions to exponential and logarithmic functions. We connect these new function families to arithmetic and geometric sequences, examine transformations, and work with real-world contexts such as growth and decay. A major emphasis includes the use of logarithms as inverses of exponentials.

2.1 Change in Arithmetic and Geometric Sequences

Arithmetic sequence: A sequence with a constant difference \(d\). If the first term is \(a_1\), the \(n\)-th term is \(\displaystyle a_n = a_1 + (n-1)d\).

Geometric sequence: A sequence with a constant ratio \(r\). If the first term is \(a_1\), the \(n\)-th term is \(\displaystyle a_n = a_1 \cdot r^{n-1}\).

These sequences illustrate linear vs. exponential patterns of change, a foundation for understanding linear vs. exponential functions.

2.2 Change in Linear and Exponential Functions

Linear function: \(f(x) = mx + b\), with constant rate of change \(m\). Exponential function: \(f(x) = a \cdot b^x\), with a multiplicative rate of change from one \(x\)-value to the next.

2.3 Exponential Functions

An exponential function is typically written as: \[ f(x) = A \cdot r^x, \] where \(A\) is an initial value and \(r\) is the growth (or decay) factor. If \(r > 1\), we have growth; if \(0 < r < 1\), we have decay. Continuous growth is often modeled with \(f(x) = A e^{kx}\), where \(k>0\) is the growth constant.

2.4 Exponential Function Manipulation

Common manipulations include rewriting \(b^x\) in terms of \(e\). For example, if \(b = e^k\), then \(b^x = e^{kx}\). Also, exponent laws let us transform sums in exponents into products: \[ b^{x+y} = b^x \cdot b^y, \quad (b^x)^m = b^{xm}. \]

2.5 Exponential Function Context and Data Modeling

Applications of exponential functions are abundant in finance (compound interest), population modeling (growth or decline), and certain physical processes (radioactive decay, carbon dating, cooling curves). Key formula examples:

2.6 Competing Function Model Validation

When deciding if data fits better with a linear model \(f(x) = mx + b\) or an exponential model \(f(x) = A \cdot r^x\), we often measure how well each model predicts data points (e.g., using residuals or correlation measures). We then validate which function more closely represents the trend.

2.7 Composition of Functions

The composition \( (f \circ g)(x) = f(g(x)) \) is especially important with exponentials and logarithms. For instance, if \(f(x) = \ln(x)\) and \(g(x) = e^x\), then \(f(g(x)) = \ln(e^x) = x\). This underscores their inverse relationship.

2.8 Inverse Functions

A function \(f(x)\) has an inverse \(f^{-1}(x)\) if and only if \(f\) is one-to-one (bijective in its domain-range pairing). The line \(y = x\) acts as a reflection line between \(f\) and \(f^{-1}\). For exponentials, the inverse is logarithmic.

2.9 Logarithmic Expressions

A logarithm is the inverse of an exponential. If \(\displaystyle y = b^x\), then \(\displaystyle x = \log_b(y)\). Key properties:

2.10 Inverses of Exponential Functions

Exponential functions \( f(x) = a \cdot r^x \) have inverse \( f^{-1}(x) = \log_r\left(\frac{x}{a}\right)\). For the natural exponential \( e^x \), the inverse is \( \ln(x) \).

2.11 Logarithmic Functions

A general logarithmic function can be written: \[ g(x) = A + B \log_r(x - C), \] involving transformations of the base \(\log_r\). Logarithmic functions feature vertical asymptotes where the argument is zero, i.e., \(x - C = 0\).

2.12 Logarithmic Function Manipulation

Using log rules, we can convert products into sums, quotients into differences, and powers into multipliers. This is invaluable for solving equations that combine multiple exponential terms.

2.13 Exponential and Logarithmic Equations and Inequalities

To solve exponential equations like \(2^x = 7\), we take a logarithm (often natural or base 2): \[ x = \log_2(7). \] For logarithmic equations like \(\ln(x) = 3\), we exponentiate: \[ x = e^3. \] Inequalities follow similar transformations, but we must monitor sign changes or domain constraints carefully.

2.14 Logarithmic Function Context and Data Modeling

Logarithmic scaling is used in measuring sound intensity (decibels), pH in chemistry, the Richter scale for earthquakes, and more. Recognizing when data suggests a logarithmic pattern is a key skill in modeling real-world scenarios.

2.15 Semi-log Plots

A semi-log plot is a graph where one axis (often the y-axis) is on a logarithmic scale, transforming an exponential relationship into a linear one. This technique is crucial in fields like biology (bacterial growth), finance (investment growth), and physics (radioactive decay).

Unit 3: Trigonometric and Polar Functions

Unit 3 introduces periodic functions and extends into trigonometric and polar representations. We examine the sine, cosine, and tangent functions, their transformations, and the translation of polar coordinates into Cartesian coordinates. We also address inverse trigonometric functions and how to analyze rates of change in polar contexts.

3.1 Periodic Phenomena

Periodic functions repeat their values in regular intervals, known as periods. For instance, \(\sin(\theta + 2\pi) = \sin(\theta)\). Modeling waves, circular motion, and seasonal changes often relies on periodic functions.

3.2 Sine, Cosine, and Tangent

These fundamental trigonometric functions can be defined using right triangles or the unit circle, with angles measured in degrees or radians.

• \(\sin(\theta)\): y-coordinate on the unit circle
• \(\cos(\theta)\): x-coordinate on the unit circle
• \(\tan(\theta) = \sin(\theta)/\cos(\theta)\)

3.3 Sine and Cosine Function Values

Special angles at \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \dots\) produce well-known sine and cosine values. For instance, \(\sin(\frac{\pi}{6}) = \frac{1}{2}\), \(\cos(\frac{\pi}{3}) = \frac{1}{2}\).

3.4 Sine and Cosine Function Graphs

Sine and cosine functions oscillate between -1 and 1, with standard period \(2\pi\). Graphs feature maxima, minima, and zeros in a repeating pattern. Key points for one period can be plotted to shape the entire wave.

3.5 Sinusoidal Functions

A general sinusoidal function can be written as: \[ y = A \sin(B(x - C)) + D \quad \text{or} \quad y = A \cos(B(x - C)) + D, \] where \(A\) is amplitude, \(\frac{2\pi}{B}\) is the period, \(C\) is the horizontal shift (phase shift), and \(D\) is the vertical shift.

3.6 Sinusoidal Function Transformations

Similar to transformations in other functions, \((x - C)\) inside the argument shifts the graph horizontally by \(C\), multiplying \(x\) by \(B\) changes the period, multiplying the function by \(A\) scales the amplitude, and adding \(D\) shifts the function vertically.

3.7 Sinusoidal Function Context and Data Modeling

Sinusoidal models often describe waves, oscillations, or cyclical phenomena—like seasonal temperatures, daily tides, or AC voltage. By identifying amplitude, period, phase shift, and vertical shift, we tailor a sinusoid to real-world data.

3.8 The Tangent Function

Tangent has a period of \(\pi\) and vertical asymptotes where \(\cos(\theta) = 0\). It can also be represented in transformations: \[ y = A \tan(B(x - C)) + D. \]

3.9 Inverse Trigonometric Functions

\(\sin^{-1}(y)\), \(\cos^{-1}(y)\), and \(\tan^{-1}(y)\) are principal-value functions that return angles. They are inverses restricted to specific ranges to ensure one-to-one behavior. For example, \(\sin^{-1}(x)\) typically has a range of \([- \frac{\pi}{2}, \frac{\pi}{2}]\).

3.10 Trigonometric Equations and Inequalities

Solving trig equations like \(\sin(x) = \frac{\sqrt{2}}{2}\) involves identifying all angles that produce the same function value within the domain of interest. Inequalities (e.g., \(\cos x > 0\)) can be solved by analyzing intervals on the unit circle or sine/cosine graphs.

3.11 The Secant, Cosecant, and Cotangent Functions

These are reciprocals: \[ \sec(x) = \frac{1}{\cos(x)}, \quad \csc(x) = \frac{1}{\sin(x)}, \quad \cot(x) = \frac{1}{\tan(x)}. \] Each has periodic asymptotes where the denominator is zero. They share transformation properties analogous to sine, cosine, and tangent.

3.12 Equivalent Representations of Trigonometric Functions

Using trigonometric identities (Pythagorean, sum/difference, double-angle), we can rewrite trigonometric expressions in various forms. This is important for simplifying, solving equations, and matching certain problem requirements.

3.13 Trigonometry and Polar Coordinates

Polar coordinates \((r, \theta)\) relate to Cartesian \((x, y)\) via: \[ x = r \cos(\theta), \quad y = r \sin(\theta). \] Trigonometric functions naturally describe circular/spiral motion, making polar coordinates efficient for certain contexts.

3.14 Polar Function Graphs

A polar function is given by \(r = f(\theta)\). Examples include cardioids, lemniscates, and roses. Graphing these functions involves computing \((r, \theta)\) points and translating them into \((x, y)\) if needed. Rates of change in polar functions can reveal tangential velocity or arc length features.

3.15 Rates of Change in Polar Functions

Differentiating polar functions with respect to \(\theta\) requires specialized formulas if we want, say, the slope in Cartesian coordinates. One must use \[ x(\theta) = r(\theta)\cos(\theta), \quad y(\theta) = r(\theta)\sin(\theta). \] Then \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}. \] This allows us to find tangents and examine local rates of change.

Unit 4: Functions Involving Parameters, Vectors, and Matrices

The final unit addresses parametric functions, vectors, matrix operations, and their roles in modeling. We explore how parametric definitions extend traditional function concepts, how vectors represent magnitude and direction, and how matrices act as functions for linear transformations. These ideas are pivotal in bridging the gap between high school mathematics and more advanced fields in engineering and applied mathematics.

4.1 Parametric Functions

A parametric function defines both \(x\) and \(y\) (or more variables) in terms of a parameter \(t\). For instance, \[ x = x(t), \quad y = y(t). \] This allows describing curves that are not expressible as \(y = f(x)\) in a single rectangular form.

4.2 Parametric Functions Modeling Planar Motion

In physics or robotics, parametric equations track an object’s motion over time: \[ x(t) = v_x t + x_0, \quad y(t) = v_y t + y_0, \] for constant velocity in 2D. With acceleration, we may have quadratic or higher-degree terms.

4.3 Parametric Functions and Rates of Change

Using parametric derivatives: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}, \] we can find slopes of tangent lines or other rates of change in parametric systems. The second derivative \(\frac{d^2y}{dx^2}\) is similarly found by differentiating \(\frac{dy}{dx}\) with respect to \(x\).

4.4 Parametrically Defined Circles and Lines

A circle of radius \(r\) can be parametrized as \[ x(t) = r \cos(t), \quad y(t) = r \sin(t). \] A line with slope \(m\) and intercept \(b\) can be parametrized as \[ x(t) = t, \quad y(t) = m t + b. \] Parametric forms allow flexible descriptions and are especially helpful in multi-constraint problems.

4.5 Implicitly Defined Functions

An equation \(F(x, y) = 0\) defines \(y\) implicitly in terms of \(x\). Such relations may not yield a neat “\(y = \dots\)” expression. Parametric or polar coordinates can sometimes circumvent the difficulty by representing the same curve in a more tractable way.

4.6 Conic Sections

Conic sections (circle, ellipse, parabola, hyperbola) can be defined implicitly (\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)), or parametrically. For example, an ellipse can be parametrized as \[ x(t) = a \cos(t), \quad y(t) = b \sin(t). \] Understanding these forms is core to advanced geometry.

4.7 Parametrization of Implicitly Defined Functions

Parametrization can help solve or interpret complex curves:

• Eliminate complicated radical forms or solve for multiple branches of a relation.
• Analyze rates of change or motion restricted by certain constraints.

4.8 Vectors

A vector in the plane can be written as \(\mathbf{v} = \langle v_x, v_y \rangle\). Key properties and operations:

4.9 Vector-Valued Functions

A vector-valued function \(\mathbf{r}(t)\) might be \(\langle x(t), y(t)\rangle\). Differentiation or integration of \(\mathbf{r}(t)\) yields velocity and displacement vectors. This is common in advanced physics and engineering contexts.

4.10 Matrices

A matrix is a rectangular array of numbers. If a matrix is \(2 \times 2\), for example \[ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \] we can use it for transformations, solving systems of equations, etc.

4.11 The Inverse and Determinant of a Matrix

The determinant of a \(2 \times 2\) matrix is \(\det(M) = ad - bc\). If \(\det(M) \neq 0\), the matrix is invertible, and \[ M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. \] Larger matrices involve more complex determinant formulas and methods (like row reduction) for finding inverses.

4.12 Linear Transformations and Matrices

A linear transformation \(\mathbf{T}: \mathbb{R}^2 \to \mathbb{R}^2\) can be represented by multiplying a matrix \(M\) by a vector \(\mathbf{x}\). For instance, \[ \mathbf{T}(\mathbf{x}) = M \mathbf{x}. \] The columns of \(M\) are the images of the basis vectors under transformation.

4.13 Matrices as Functions

We can interpret matrix multiplication as applying a function to a vector. Each input vector \(\mathbf{x}\) yields an output vector \(\mathbf{T}(\mathbf{x})\). This viewpoint is central to linear algebra, bridging function concepts in precalculus with more advanced matrix-based methods.

4.14 Matrices Modeling Contexts

Matrices model systems of linear equations, transformations, Markov chains, computer graphics transformations, and more. The concept of using a matrix function to transform data is widely used across science and engineering.

Conclusion: Unifying Concepts in AP Precalculus

AP Precalculus merges multiple domains—polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vectors, and matrices—into a cohesive prelude to calculus and higher-level mathematics. Each unit builds on the prior, emphasizing rates of change, transformations, and modeling real-world phenomena.

In Unit 1, we examined polynomial and rational function behavior, focusing on zeros, asymptotes, and end behavior as well as transformations and modeling. In Unit 2, exponential and logarithmic functions took center stage, revealing a multiplicative perspective on growth and decay, along with logarithms as inverses and data modeling tools. Unit 3 introduced periodic phenomena through trigonometric functions and extended to polar coordinates. Finally, Unit 4 broadened our concept of a function to parametric, vector, and matrix-based models, integrating advanced tools that unify geometry, algebra, and real-world applications.

These areas serve as stepping stones to calculus and beyond: understanding rates of change conceptually sets the stage for derivatives, while transformations, inverses, and composition pave the way for advanced function analysis. Mastery of these AP Precalculus formulae and concepts provides a robust mathematical foundation, readying students for the challenges of college-level STEM courses.

Keep this guide as a reference as you explore more nuanced problems and applications. Engage in active problem-solving, practice with real-world data modeling, and explore how each function family can describe phenomena in science, finance, engineering, and beyond. AP Precalculus is more than a collection of formulas—it’s a framework for interpreting and navigating the mathematical tapestry of our world.