Math · Pre-Calculus

Pre-Calculus, explained unit by unit

Q: What do you learn in Pre-Calculus?

Polynomial and rational functions, exponential and logarithmic functions and models, trigonometric and polar functions, and functions involving parameters, vectors and matrices. In Mathflow, that’s 4 units and 32 skills.

Q: Is Pre-Calculus the same as AP Precalculus?

Not always: Pre-Calculus courses vary from school to school. Mathflow’s four units follow the four topics of the College Board’s AP Precalculus course framework. The AP exam assesses the first three units.

Q: What grade is Pre-Calculus?

Pre-Calculus usually comes after Algebra 2, often in 11th or 12th grade, and before Calculus.

Q: Is Pre-Calculus hard?

It asks you to handle many function families at once and to move between equations, graphs, tables and real data. Solid Algebra 2 skills, such as factoring, logarithms and the unit circle, make it much easier.

Q: Do I need Pre-Calculus before Calculus?

In most schools, yes. Calculus uses every function family from Pre-Calculus, and its limits build on behavior you study here, such as asymptotes and end behavior.

Q: How can I practice Pre-Calculus at home?

Redo the worked problems on this page without looking, then practice skill by skill. Mathflow covers the whole course in three levels per skill, with Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

Rational, exponential, logarithmic, trigonometric and polar functions, then parameters, vectors and matrices: here is everything Pre-Calculus covers, with worked problems and the mistakes to avoid. Then practice each skill in Mathflow, from intuition to solving.

Start for free See the 4 units

On the unit circle, the point at the angle

\dfrac{\pi}{3}

\left(\cos\dfrac{\pi}{3}, \sin\dfrac{\pi}{3}\right) = \left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)

What you’ll learn

The 4 units of Pre-Calculus

The course as it is organized in Mathflow, in order. Each unit is split into precise skills that you practice one at a time.

Unit 17 skills
Polynomial & Rational Functions
Compare how quantities change together, link the zeros of a polynomial to its whole graph, build the full profile of a rational function, and choose a model with clear assumptions.
- Compare how quantities change together
- Use finite differences to investigate polynomial change
- Connect polynomial roots with whole-graph behavior
- Build a complete rational-function behavior profile
- Choose an equivalent form that exposes function behavior
- Track function transformations across representations
- Select and construct a function model with stated assumptions
Unit 28 skills
Exponential & Logarithmic Modeling
Connect sequences with linear and exponential models, compare models with residuals, compose and invert functions, and solve and interpret logarithmic models.
- Connect sequences with linear and exponential models
- Fit and reinterpret exponential growth parameters
- Compare competing models using residual evidence
- Compose functions and determine the valid inputs
- Construct inverses with appropriate domain restrictions
- Connect logarithms to inverse graphs and equivalent forms
- Solve exponential and logarithmic inequalities with domain checks
- Interpret logarithmic models and semi-logarithmic displays
Unit 39 skills
Trigonometry & Polar Functions
Link circular motion and radians to periodic change, build sinusoidal models, solve trigonometric equations, prove identities, and sketch polar curves.
- Connect circular motion, radian measure, and periodic change
- Construct sinusoidal models from graphs or measurements
- Analyze tangent and reciprocal trigonometric graphs
- Interpret principal inverse trigonometric values
- Solve trigonometric equations and inequalities on a domain
- Use trigonometric identities to prove equivalence or solve
- Translate between polar and Cartesian descriptions
- Sketch polar functions by tracking angle and radius
- Describe radial change without using derivatives
Unit 48 skills
Parameters & Linear Algebra
Describe paths with parametric equations, study conic sections, combine vectors, and use matrices to transform the plane, measure areas and model transitions between states.
- Model a planar path with parametric coordinate functions
- Analyze implicit relations and conic sections
- Parametrize an implicitly defined curve
- Use planar vectors to combine directions and magnitudes
- Read a vector-valued function as a changing quantity
- Connect matrix operations, invertibility, and geometric area
- Represent and compose planar linear transformations
- Model transitions between states with matrices

Pre-Calculus at a glance

Pre-Calculus usually comes after Algebra 2 and before Calculus. It takes the function families you already know and studies them as a whole: how they change, how they combine, and how they model real data.

Before you start. You need solid Algebra 2: factoring and polynomial division, exponents and logarithms, and the trigonometry of right triangles and the unit circle. Geometry helps too, for conics and vectors.

The four units. They follow the same four topics as the AP Precalculus course framework of the College Board. Its exam assesses the first three units; the fourth, on parameters, vectors and matrices, is not on the exam.

In Mathflow. The course has $4$ units and $32$ skills. Each skill is a series in three levels, and each unit ends with a Chapter Challenge.

Worked problems

Two problems, solved step by step

The profile of a rational function and a trigonometric equation: two classics of Pre-Calculus. Try each one on paper before you read the solution.

Problem 1
Profile of a rational function
Let $f(x) = \dfrac{x^2 - 4}{x^2 - x - 6}$ . Find its domain, its zeros, its hole, its asymptotes and its $y$ -intercept.
Solution
Factor the numerator and the denominator:
$f(x) = \dfrac{(x - 2)(x + 2)}{(x - 3)(x + 2)}$
Domain. The denominator is $0$ at $x = 3$ and $x = -2$ , so the domain is every real number except $3$ and $-2$ .
Hole. The factor $x + 2$ cancels, so $f(x) = \dfrac{x - 2}{x - 3}$ for $x \neq -2$ . At $x = -2$ the graph has a hole at $\left(-2, \dfrac{4}{5}\right)$ , because $\dfrac{-2 - 2}{-2 - 3} = \dfrac{4}{5}$ .
Zero. $x = 2$ , where the remaining numerator is $0$ .
Vertical asymptote. $x = 3$ . Just to the right of $3$ , $f(x)$ goes up without bound; just to the left, it goes down without bound.
Horizontal asymptote. The numerator and the denominator have the same degree, so the asymptote is the ratio of the leading coefficients: $y = \dfrac{1}{1} = 1$ .
$y$ -intercept. $f(0) = \dfrac{-4}{-6} = \dfrac{2}{3}$ .
Problem 2
A trigonometric equation on an interval
Solve $2\sin^2\theta - \sin\theta - 1 = 0$ for $0 \le \theta < 2\pi$ .
Solution
The equation is a quadratic in $\sin\theta$ . Factor it as you would factor $2s^2 - s - 1$ :
$(2\sin\theta + 1)(\sin\theta - 1) = 0$
So $\sin\theta = 1$ or $\sin\theta = -\dfrac{1}{2}$ .
Case 1. On $[0, 2\pi)$ , $\sin\theta = 1$ only at $\theta = \dfrac{\pi}{2}$ .
Case 2. The reference angle of $\sin\theta = \dfrac{1}{2}$ is $\dfrac{\pi}{6}$ . The sine is negative in quadrants III and IV, so $\theta = \pi + \dfrac{\pi}{6} = \dfrac{7\pi}{6}$ or $\theta = 2\pi - \dfrac{\pi}{6} = \dfrac{11\pi}{6}$ .
The solutions are $\dfrac{\pi}{2}$ , $\dfrac{7\pi}{6}$ and $\dfrac{11\pi}{6}$ . Check: for $\theta = \dfrac{7\pi}{6}$ , $2 \cdot \dfrac{1}{4} + \dfrac{1}{2} - 1 = 0$ .

How Mathflow teaches it

Pre-Calculus in three levels

Every skill of the course is a series of short missions in three levels. Here is what you actually do in three of them.

Level 1 · Explore
Explore and play with the concepts to build your intuition.
Level 2 · Explain
Turn your intuitions into the right words and uncover hidden mistakes.
Level 3 · Solve
Reason it out, crunch the numbers, and find your way to the answer.

You don’t start with a page of formulas. In the first level, you act on the math: you change a power, drag a curve onto measured data, resize a parallelogram and watch the function respond. In the second level, you put words on what you saw: you decide which of two students is right, find the first wrong line of a solution, or explain why a graph looks the way it does. In the third level, you solve problems on your own, like the ones you meet in class and on tests.

If you get stuck, Alphi, the AI math tutor, gives you a hint instead of the answer, at any hour. When the skills of a unit are done, a Chapter Challenge checks the whole unit.

Unit 1 · Polynomial & Rational Functions
1Build a complete rational-function behavior profile
- Level 1 · Explore
  Tap where the graph of $f(x) = \dfrac{(x - 1)(x + 2)}{(x - 1)(x - 3)}$ should show a hole. Then change the power $n$ in $f(x) = \dfrac{x + 1}{(x - 2)^n}$ until both sides of $x = 2$ go up.
- Level 2 · Explain
  Noah graphed $f(x) = \dfrac{x + 9}{(x - 1)(x + 2)}$ in a small window and saw no zero, and you explain why. You also tap the first wrong line in Grace’s profile of $f(x) = \dfrac{x^2 + 2x}{x^2 - 4}$ .
- Level 3 · Solve
  You find the coordinates of the hole in the graph of $g(x) = \dfrac{x^2 + x - 2}{x^2 - x}$ , and the curve that $f(x) = \dfrac{x^3 + x^2 - 4}{x}$ approaches at both ends.
Unit 3 · Trigonometry & Polar Functions
2Construct sinusoidal models from graphs or measurements
- Level 1 · Explore
  Drag two lines to the highest and the lowest measured depths at a dock, then drag the highest point of the curve until the model fits the measurements.
- Level 2 · Explain
  The temperature is lowest, $58^\circ\text{F}$ , at 4 in the morning and highest, $74^\circ\text{F}$ , at 4 in the afternoon. You tap the first wrong line of Jordan’s model, and you decide whether a model fitted to one April day can predict the temperature $2{,}000$ hours later.
- Level 3 · Solve
  Average temperatures peak at $78^\circ\text{F}$ in July and are lowest, $38^\circ\text{F}$ , in January. You complete the sinusoidal model.
Unit 4 · Parameters & Linear Algebra
3Connect matrix operations, invertibility, and geometric area
- Level 1 · Explore
  Drag $Q$ until the parallelogram built on the two columns of a matrix has an area of $6$ , and set $k$ so that a matrix has no inverse.
- Level 2 · Explain
  You tap the first wrong line in Maya’s determinant of $\left[\begin{smallmatrix} 4 & -3 \\ 2 & 5 \end{smallmatrix}\right]$ , and you explain why a matrix that sends the whole plane onto a line has no inverse.
- Level 3 · Solve
  A figure with an area of $7$ is mapped by $\left[\begin{smallmatrix} 1 & 3 \\ 2 & 1 \end{smallmatrix}\right]$ . You find the area of its image.

Practice Pre-Calculus for free

Common mistakes

Five mistakes to avoid

These slips come up again and again in Pre-Calculus. Learn to spot them, and you’ll catch them in your own work.

Forgetting the hole after canceling
$\dfrac{x^2 - 4}{x - 2}$ simplifies to $x + 2$ , but only for $x \neq 2$ . Its graph is the line $y = x + 2$ with a hole at $(2, 4)$ : canceling never adds $2$ back to the domain.
Reading $f^{-1}$ as a reciprocal
$f^{-1}$ is the inverse function, not $\dfrac{1}{f}$ . For $f(x) = 2x$ , the inverse is $f^{-1}(x) = \dfrac{x}{2}$ , while $\dfrac{1}{f(x)} = \dfrac{1}{2x}$ .
Composing in the wrong order
$(f \circ g)(x) = f(g(x))$ : apply $g$ first. With $f(x) = x^2$ and $g(x) = x + 1$ , you get $f(g(2)) = 9$ but $g(f(2)) = 5$ .
Keeping only the calculator’s angle
$\sin^{-1}(0.5)$ returns only $\dfrac{\pi}{6}$ . On $[0, 2\pi)$ , the equation $\sin\theta = 0.5$ also has the solution $\dfrac{5\pi}{6}$ . Use the reference angle and the quadrants to find every solution.
Picking the wrong quadrant in polar form
For the point $(-1, 1)$ , $\tan\theta = -1$ , but $\theta = -\dfrac{\pi}{4}$ points into quadrant IV. The point is in quadrant II, so $\theta = \dfrac{3\pi}{4}$ , with $r = \sqrt{2}$ .

Frequently asked questions

What do you learn in Pre-Calculus?

Polynomial and rational functions, exponential and logarithmic functions and models, trigonometric and polar functions, and functions involving parameters, vectors and matrices. In Mathflow, that’s 4 units and 32 skills.

Is Pre-Calculus the same as AP Precalculus?

Not always: Pre-Calculus courses vary from school to school. Mathflow’s four units follow the four topics of the College Board’s AP Precalculus course framework. The AP exam assesses the first three units.

What grade is Pre-Calculus?

Pre-Calculus usually comes after Algebra 2, often in 11th or 12th grade, and before Calculus.

Is Pre-Calculus hard?

It asks you to handle many function families at once and to move between equations, graphs, tables and real data. Solid Algebra 2 skills, such as factoring, logarithms and the unit circle, make it much easier.

Do I need Pre-Calculus before Calculus?

In most schools, yes. Calculus uses every function family from Pre-Calculus, and its limits build on behavior you study here, such as asymptotes and end behavior.

How can I practice Pre-Calculus at home?

Redo the worked problems on this page without looking, then practice skill by skill. Mathflow covers the whole course in three levels per skill, with Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

Start for free

Standards

Pre-Calculus and the standards

The College Board’s AP Precalculus course has four units: polynomial and rational functions; exponential and logarithmic functions; trigonometric and polar functions; and functions involving parameters, vectors and matrices. Units 1 to 3 are assessed on the AP exam, and unit 4 is not.

The Common Core State Standards don’t define a Pre-Calculus course. Appendix A of the standards lists Precalculus among the courses that can follow Algebra II, and places topics such as vectors, matrices and inverse trigonometric functions in standards marked (+), beyond the three core courses.

Pre-Calculus, explained unit by unit

Polynomial & Rational Functions

Exponential & Logarithmic Modeling

Trigonometry & Polar Functions

Parameters & Linear Algebra

Pre-Calculus at a glance

Profile of a rational function

A trigonometric equation on an interval

Level 1 · Explore

Level 2 · Explain

Level 3 · Solve

1Build a complete rational-function behavior profile

2Construct sinusoidal models from graphs or measurements

3Connect matrix operations, invertibility, and geometric area

Forgetting the hole after canceling

Reading f−1f^{-1}f−1 as a reciprocal

Composing in the wrong order

Keeping only the calculator’s angle

Picking the wrong quadrant in polar form

Math is all about flow.

Reading $f^{-1}$ as a reciprocal