Precalculus reference sheet

This sheet follows the ten units of the course in order. Each unit starts with the big idea in a few sentences, then gives the definitions, formulas and rules you will look up most often, with the conditions they need. When you want the full explanation or more examples, follow the lesson links under each topic.

Angles are in radians unless they are marked in degrees. A formula holds wherever both sides are defined unless a condition says otherwise.

Functions, Relations, and Graphs

A function assigns exactly one output to each input. This unit sets up the language for talking about functions, from intervals and domains to graphs and rates of change. It then builds new functions from old ones by shifting, combining, composing and inverting them.

Sets and intervals

A set is a collection of distinct elements, and $x \in A$ means that $x$ belongs to $A$. The empty set $\varnothing$ has no elements, so it is not the same as $\{0\}$, which has one. Set-builder notation describes a set by a condition, as in $\{x \mid x > 2\}$.

Infinity is not a number, so it always gets a parenthesis. The union $A \cup B$ holds the elements in $A$ or $B$ or both, and the intersection $A \cap B$ holds the elements in both. For example $[0, 5) \cap (2, 8] = (2, 5)$.

Adding the same number to both sides of an inequality, or multiplying both sides by the same positive number, keeps its direction. Multiplying or dividing by a negative number reverses it, so $-2x < 6$ gives $x > -3$. A compound inequality such as $-1 \le 2x + 3 < 7$ is solved on all three parts at once, which gives $-2 \le x < 2$. Testing one value from your answer in the original inequality catches a missed reversal.

Lessons Sets, intervals and inequalities

Functions, domain and range

A relation is a set of ordered pairs $(x, y)$. It is a function when each input $x$ is paired with exactly one output $y$, written $y = f(x)$. The input is the independent variable and the output is the dependent variable. Several inputs may share an output, but one input may not have two. A curve in the plane is the graph of a function of $x$ exactly when no vertical line meets it more than once.

• Notation. The symbol $f(x)$ is the output at the input $x$, not $f$ times $x$. To find $f(a + h)$, replace every $x$ with $a + h$. For $f(x) = x^2$ this gives $a^2 + 2ah + h^2$, which in general is neither $f(a) + h$ nor $f(a) + f(h)$.
• Natural domain. When only a formula is given, the domain is every real $x$ for which the formula gives a real number. Leave out zeros of denominators, keep the radicand of an even root at least $0$, and keep the argument of a logarithm positive. Odd roots accept every real number. A domain can also be stated outright or set by the context, such as a length that must be positive.
• Range. The range is the set of outputs the function actually takes. It sits inside the codomain, the set where outputs are allowed to lie. On a graph it is the projection of the graph onto the $y$-axis.

For example the function

has domain $[-3, 1) \cup (1, \infty)$. Restrictions in the original formula stay even after simplifying, so $\frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$ although it simplifies to $x + 2$. The function $x^2$ has range $[0, \infty)$ on all of $\mathbb{R}$, but range $[0, 4]$ when its domain is $[-1, 2]$.

Lessons Relations and functions, Function notation, Domains, Range and codomain

Lines, graphs and rates of change

The slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 \ne x_2$, is

A linear function changes at a constant rate, its slope, measured in output units per input unit. When $b = 0$ and $m \ne 0$ the output is proportional to the input. Two distinct nonvertical lines are parallel exactly when their slopes are equal, and two nonvertical lines are perpendicular exactly when their slopes multiply to $-1$.

The average rate of change of $f$ from $x = a$ to $x = b$, with $a \ne b$, is

the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. For $f(x) = x^2$ from $1$ to $3$ it is $\frac{9 - 1}{3 - 1} = 4$. Its sign says whether the output rose or fell overall.

To read a graph, the $x$-intercepts are the solutions of $f(x) = 0$ and the $y$-intercept is $f(0)$ when $0$ is in the domain. A function is increasing on an interval when $x_1 < x_2$ gives $f(x_1) < f(x_2)$ there, decreasing when it gives $f(x_1) > f(x_2)$, and constant when it gives $f(x_1) = f(x_2)$. A local maximum value $f(c)$ is at least as large as $f(x)$ for every $x$ near $c$, and a global maximum value is at least as large as every value of $f$. Minimum values work the same way with the inequality reversed. Values read from a graph are estimates until the formula confirms them.

Lessons Linear functions, Reading graphs, Average rate of change

Piecewise, absolute value and symmetric functions

A piecewise function uses different formulas on different parts of its domain. The branch conditions should not overlap, and an input that meets none of them is not in the domain. On the graph a closed dot marks an endpoint that belongs to its branch and an open dot marks one that does not. The absolute value is the piecewise function

which measures the distance from $x$ to $0$. In the same way $\lvert x - a \rvert$ is the distance from $x$ to $a$. For $c > 0$,

The same holds with $\le$ and $\ge$, endpoints included. If $c = 0$, the equation $\lvert u \rvert = 0$ means $u = 0$, the inequality $\lvert u \rvert < 0$ has no solutions and $\lvert u \rvert > 0$ holds for every $u \ne 0$. If $c < 0$, the equation $\lvert u \rvert = c$ and the inequality $\lvert u \rvert < c$ have no solutions, while $\lvert u \rvert > c$ always holds. When the right side contains the variable, as in $\lvert x - 1 \rvert = 2x$, check each candidate in the original equation. There the candidates are $-1$ and $\frac{1}{3}$, and only $\frac{1}{3}$ works.

A function is even when $f(-x) = f(x)$ and odd when $f(-x) = -f(x)$ for every $x$ in its domain, which must contain $-x$ whenever it contains $x$. An even graph is symmetric about the $y$-axis and an odd graph is symmetric about the origin. For example $x^2$ is even, $x^3$ is odd and $x^2 + x$ is neither. The zero function is both.

Lessons Piecewise functions, Absolute value, Even and odd functions

Transformations of graphs

Start from the graph of $y = f(x)$ and let $c > 0$.

Changes inside the function act on $x$ and work the opposite way to how they look. In general, for nonzero $a$ and $b$ the graph of $y = a\,f\big(b(x - h)\big) + k$ takes each point $(p, q)$ of the graph of $f$ to

For example $y = 2\sqrt{x - 3} + 1$ takes the point $(4, 2)$ of $y = \sqrt{x}$ to $(7, 5)$.

Lessons Shifts and reflections, Stretches and compressions

Combining and inverting functions

A composition applies $g$ first, and the order matters. With $f(x) = \sqrt{x}$ and $g(x) = x - 2$, the composition $f(g(x)) = \sqrt{x - 2}$ has domain $[2, \infty)$, while $g(f(x)) = \sqrt{x} - 2$ has domain $[0, \infty)$.

A function is one-to-one when different inputs always give different outputs, so $f(x_1) = f(x_2)$ forces $x_1 = x_2$. Its graph passes the horizontal line test, and a function that is strictly increasing or strictly decreasing is always one-to-one. A one-to-one $f$ has an inverse $f^{-1}$ defined by

Then $f^{-1}(f(x)) = x$ for every $x$ in the domain of $f$ and $f(f^{-1}(y)) = y$ for every $y$ in the range of $f$. Domain and range swap, and the graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$. To find a formula, solve $y = f(x)$ for $x$ and then swap the letters, so $f(x) = 2x + 3$ has $f^{-1}(x) = \frac{x - 3}{2}$. The inverse is not the reciprocal, since $f^{-1}(x)$ is generally not $\frac{1}{f(x)}$. A function that is not one-to-one gets an inverse once its domain is restricted, as $x^2$ on $[0, \infty)$ has the inverse $\sqrt{x}$.

Lessons Algebra of functions, Composition, One-to-one functions, Inverse functions

Polynomial, Rational, and Algebraic Functions

Polynomials are built from constants and a variable using only addition and multiplication, and rational functions divide one polynomial by another. Their graphs are read from zeros, end behavior and asymptotes. Complex numbers make it possible to factor every polynomial completely.

Quadratics, distance and circles

A quadratic function $f(x) = ax^2 + bx + c$ with $a \ne 0$ has a parabola as its graph, opening up when $a > 0$ and down when $a < 0$. Completing the square gives the vertex form

with vertex $(h, k)$ and axis of symmetry $x = h$. For example $x^2 + 6x + 5 = (x + 3)^2 - 4$ has vertex $(-3, -4)$. The zeros of $ax^2 + bx + c$ are

With real coefficients the discriminant says what kind of zeros to expect.

For points $(x_1, y_1)$ and $(x_2, y_2)$ the distance and the midpoint are

The circle with center $(h, k)$ and radius $r > 0$ is

A general equation $x^2 + y^2 + Dx + Ey + F = 0$ is put in this form by completing the square in $x$ and in $y$. For example $x^2 + y^2 - 4x + 6y - 12 = 0$ becomes $(x - 2)^2 + (y + 3)^2 = 25$, the circle with center $(2, -3)$ and radius $5$. If the right side comes out $0$ the graph is a single point, and if it comes out negative there is no graph.

Lessons Quadratic functions, Circles, distance and midpoints

Powers, roots and equations

A power function has the form $f(x) = x^p$ for a constant $p$. For an integer $n \ge 2$, the principal root $\sqrt[n]{a}$ is the real number whose $n$th power is $a$, taking the nonnegative one when $n$ is even. Even roots need $a \ge 0$, and odd roots accept every real $a$.

On $[0, \infty)$ the functions $\sqrt[n]{x}$ and $x^n$ undo each other, and for odd $n$ they do so on all of $\mathbb{R}$.

To solve a rational equation, note the excluded values, multiply both sides by the least common denominator and solve. To solve a radical equation, isolate the radical and raise both sides to the power that removes it. Both steps can produce candidates that fail the original equation, so check every one. For example $\sqrt{x + 2} = x$ leads to $x + 2 = x^2$, so $x = 2$ or $x = -1$. Only $x = 2$ works, since $\sqrt{1} \ne -1$.

Lessons Power and root functions, Rational and radical equations

Polynomials, zeros and division

A polynomial of degree $n$ has the form $P(x) = a_n x^n + \cdots + a_1 x + a_0$ with $a_n \ne 0$. For $n \ge 1$ its end behavior follows the leading term $a_n x^n$.

• Zeros. A number $c$ is a zero of multiplicity $m$ when $(x - c)^m$ is a factor of $P(x)$ and $(x - c)^{m+1}$ is not. At a real zero of odd multiplicity the graph crosses the $x$-axis, and at one of even multiplicity it touches the axis and turns back. A multiplicity of $2$ or more flattens the graph against the axis. A polynomial of degree $n \ge 1$ has at most $n$ real zeros and at most $n - 1$ turning points.
• Division. For polynomials $P$ and $D$ with $D \ne 0$ there are unique polynomials $Q$ and $R$ with $P(x) = D(x)\,Q(x) + R(x)$, where $R = 0$ or $\deg R < \deg D$. Synthetic division is a shortcut when $D(x) = x - c$.
• Remainder and factor theorems. The remainder when $P(x)$ is divided by $x - c$ is $P(c)$. So $x - c$ is a factor of $P(x)$ exactly when $P(c) = 0$.
• Rational zeros theorem. If $P$ has integer coefficients and $a_0 \ne 0$, every rational zero, written in lowest terms as $\frac{p}{q}$, has $p$ dividing $a_0$ and $q$ dividing $a_n$. The theorem only lists candidates, and some zeros may be irrational or complex.

For example the candidates for $P(x) = x^3 - 6x^2 + 11x - 6$ are $\pm 1$, $\pm 2$, $\pm 3$ and $\pm 6$. Since $P(1) = 0$, synthetic division by $x - 1$ gives

Lessons End behavior, Zeros and multiplicity, Polynomial division, Remainder and factor theorems, Rational zeros

Complex numbers and complete factorization

The imaginary unit $i$ satisfies $i^2 = -1$, and a complex number $z = a + bi$ has real part $a$ and imaginary part $b$. Plotted as the point $(a, b)$ in the complex plane, it lies at distance $\lvert z \rvert = \sqrt{a^2 + b^2}$ from the origin.

For example

For $a > 0$ we write $\sqrt{-a} = i\sqrt{a}$, but the rule $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ fails when both numbers are negative, since $\sqrt{-1}\,\sqrt{-1} = -1$.

Fundamental theorem of algebra. Every polynomial of degree $n \ge 1$ with complex coefficients has a complex zero. As a result it factors as

and has exactly $n$ zeros counted with multiplicity. When the coefficients are real, nonreal zeros come in conjugate pairs $a \pm bi$, so $P$ factors over the reals into linear factors and irreducible quadratic factors with negative discriminant. For example $x^4 - 1 = (x - 1)(x + 1)(x^2 + 1) = (x - 1)(x + 1)(x - i)(x + i)$.

Lessons Complex numbers, The fundamental theorem of algebra

Rational functions and asymptotes

A rational function is a quotient $f(x) = \frac{P(x)}{Q(x)}$ of polynomials, with $Q$ not the zero polynomial. Its domain leaves out every zero of $Q$ in the original formula, and canceling a factor does not put that point back. Factor the top and bottom and cancel common factors.

• A zero $c$ of the denominator that cancels completely leaves a hole at $x = c$, at the height the simplified formula gives there.
• A zero $c$ of the denominator that remains gives a vertical asymptote $x = c$, where $\lvert f(x) \rvert$ grows without bound as $x$ approaches $c$ from either side.

With leading terms $a_n x^n$ on top and $b_m x^m$ below, the behavior as $x \to \infty$ and as $x \to -\infty$ is

The graph can cross a horizontal or slant asymptote, but never a vertical one. For example the function

has a hole at $\left(2, \frac{4}{3}\right)$, the vertical asymptote $x = -1$ and the horizontal asymptote $y = 1$. Division gives $\frac{x^2 + 1}{x - 1} = x + 1 + \frac{2}{x - 1}$, so the line $y = x + 1$ is a slant asymptote of that function.

To graph a rational function, find its domain, holes, intercepts and asymptotes. Then make a sign chart between the zeros and excluded values, and draw each piece approaching its asymptotes.

Lessons Rational functions, Vertical asymptotes and holes, Horizontal asymptotes, Slant asymptotes, Graphing

Polynomial and rational inequalities

1. Move every term to one side so the other side is $0$, combine into one fraction if needed, and factor.
2. Mark the critical values on a number line. These are the zeros of the numerator and of the denominator.
3. Test one point in each interval to find the sign of the expression there.
4. Keep the intervals with the sign you need. Include zeros of the numerator for $\le$ or $\ge$, and never include zeros of the denominator.

Never multiply both sides by an expression whose sign you do not know. For example $x^2 - x - 6 > 0$ factors as $(x - 3)(x + 2) > 0$, with solution $(-\infty, -2) \cup (3, \infty)$. The inequality

has solution $(-1, 3]$.

Lessons Polynomial inequalities, Rational inequalities

Exponential and Logarithmic Functions

An exponential function multiplies its output by the same factor over equal steps of input. A logarithm undoes it by returning the exponent, so every fact about exponentials has a logarithmic twin. Together they model growth, decay and quantities measured on logarithmic scales.

Exponential functions and the number e

An exponential function has the form $f(x) = a\,b^x$ with $a \ne 0$ and a base $b > 0$, $b \ne 1$. Each step of $1$ in $x$ multiplies the output by $b$, so equal steps in the input give equal ratios of outputs. For $a > 0$ it shows growth when $b > 1$ and decay when $0 < b < 1$. Its domain is $\mathbb{R}$, its range is $(0, \infty)$ when $a > 0$, and the line $y = 0$ is a horizontal asymptote. For $b > 0$ and all real $x$ and $y$,

The number $e \approx 2.71828$ is the limit

It comes from compound interest. A principal $P$ at annual rate $r$, compounded $n$ times a year, grows to $P\left(1 + \frac{r}{n}\right)^{nt}$ after $t$ years. As $n$ grows this approaches $Pe^{rt}$, which is continuous compounding. A table of values suggests the limit, but numbers alone do not prove it.

Lessons Exponential functions, The number e

Logarithms

For a base $b > 0$ with $b \ne 1$ and an argument $x > 0$,

A logarithm is an exponent. The common logarithm is $\log x = \log_{10} x$ and the natural logarithm is $\ln x = \log_e x$. The values $\log_b 1 = 0$ and $\log_b b = 1$ come from $b^0 = 1$ and $b^1 = b$. The inverse rules are $\log_b(b^x) = x$ for every real $x$ and $b^{\log_b x} = x$ for every $x > 0$.

The two graphs are reflections of each other across the line $y = x$. Shifts and stretches work as for any function, so $y = \ln(x - 2) + 1$ has domain $(2, \infty)$ and vertical asymptote $x = 2$.

Lessons Logarithmic functions, Graphs

Properties of logarithms

For a base $b > 0$ with $b \ne 1$, numbers $M > 0$ and $N > 0$, and any real $p$,

There is no rule for $\log_b(M + N)$. Watch the conditions when you expand. The expression $\ln(x^2)$ is defined for every $x \ne 0$ and equals $2\ln\lvert x \rvert$, while $2\ln x$ needs $x > 0$. Change of base gives $\log_2 10 = \frac{\ln 10}{\ln 2} \approx 3.3219$. Keep exact forms like $\frac{\ln 10}{\ln 2}$ until the last step and round only the final answer.

Lessons Product and quotient properties, Power rule and change of base

Solving exponential and logarithmic equations

• Exponential equations. Write both sides with the same base and set the exponents equal, or isolate the exponential and take a logarithm of both sides. From $3e^{2x} = 12$ you get $e^{2x} = 4$, so $x = \frac{\ln 4}{2} = \ln 2$.
• Logarithmic equations. Combine into a single logarithm, rewrite in exponential form and solve. Then keep only the candidates that make every logarithm in the original equation have a positive argument. From $\log_2 x + \log_2(x - 2) = 3$ you get $x(x - 2) = 8$, so $x = 4$ or $x = -2$, and only $x = 4$ works.

Lessons Exponential equations, Logarithmic equations

Growth, decay and models

A percent rate and a continuous rate describe the same model when $1 + r = e^k$, that is when $k = \ln(1 + r)$.

A logistic model grows almost exponentially at first and then levels off toward the carrying capacity $M$. When $0 < P_0 < M$ it grows fastest at $P = \frac{M}{2}$, reached at $t = \frac{\ln A}{k}$ (a negative time when $P_0 > \frac{M}{2}$), and the curve is symmetric about that point.

To choose a model, look at equal steps in the input. Equal differences in the output suggest a linear model and equal ratios suggest an exponential one. A residual is the observed value minus the predicted value, and residuals with a clear pattern mean the model is missing something. Predictions inside the range of the data are safer than predictions beyond it. A logarithmic scale records a constant multiple of the logarithm of a quantity, as pH records $-\log_{10}$ of the concentration of hydrogen ions. Equal steps on such a scale are equal factors in the quantity, so a drop of $1$ in pH means ten times the concentration.

Lessons Growth and decay models, Choosing a model, Logistic growth

Trigonometric Functions

Trigonometry starts with ratios of sides in right triangles and grows into functions defined for every angle by the unit circle. A full turn brings you back to the same point, so these functions repeat, which makes them the natural models for waves and oscillations.

Angles and arcs

An angle in standard position has its vertex at the origin and its initial side along the positive $x$-axis. It is positive when measured counterclockwise. A full turn is $360^\circ$, or $2\pi$ radians, so to convert degrees to radians multiply by $\frac{\pi}{180}$.

Angles that differ by whole turns, $\theta + 2k\pi$ or $\theta + 360^\circ k$ for an integer $k$, are coterminal. For a circle of radius $r$ and a central angle $\theta$ in radians,

These formulas need radians, so convert degrees first. A wheel of radius $0.3$ meters turning at $10$ radians per second moves a point on its rim at $3$ meters per second.

Lessons Angles and radians, Arc length and sector area

Right triangles and special angles

For an acute angle $\theta$ in a right triangle,

and the reciprocal ratios are $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$ and $\cot\theta = \frac{1}{\tan\theta}$. Similar triangles have the same ratios, so the ratios depend only on the angle. Angles of elevation and depression are measured from the horizontal.

The isosceles right triangle with legs $1$ has hypotenuse $\sqrt{2}$. The triangle with angles $30^\circ$, $60^\circ$ and $90^\circ$ has sides $1$, $\sqrt{3}$ and $2$ opposite them. These give the exact values below.

Lessons Right triangle trigonometry, Special right triangles

Any angle and the unit circle

If $(x, y)$ is a point other than the origin on the terminal side of $\theta$ and $r = \sqrt{x^2 + y^2}$, then

On the unit circle $r = 1$, so the point at angle $\theta$ is $(\cos\theta, \sin\theta)$. You reach it from $(1, 0)$ by traveling a distance $\lvert \theta \rvert$ along the circle, counterclockwise when $\theta > 0$ and clockwise when $\theta < 0$. A full turn returns to the same point, so $\sin(\theta + 2\pi) = \sin\theta$ and $\cos(\theta + 2\pi) = \cos\theta$.

The reference angle is the acute angle between the terminal side and the $x$-axis. A function’s value at $\theta$ equals its value at the reference angle, with the sign it has in that quadrant.