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Precalculus: Build the Toolkit for Calculus, One Question at a Time
This course covers the 28 topics that bridge algebra and calculus: functions and their graphs, polynomial and rational functions, exponential and logarithmic functions, the full sweep of trigonometry (unit circle, identities, equations, Law of Sines/Cosines), systems of equations and matrices, sequences and series, the binomial theorem, vectors, complex numbers, conic sections, polar coordinates, and parametric equations. It stops right before derivatives and integrals, so every topic here is something you'll actually use once calculus starts.
A lesson starts from what you already know and moves forward one question at a time. If you're asked to graph a transformed function, the tutor doesn't show you the transformation rules first — it asks what happens to a point when you add 3 inside the parentheses, waits for your answer, and builds from there. Wrong turns get noticed and corrected on the spot; the tutor asks a follow-up that targets exactly the gap instead of re-explaining everything.
Who it is for
- Students who finished Algebra II and want to get ready for calculus
- Students currently in a precalculus class who want to talk through topics they're stuck on
- Homeschool students who need a full precalculus course with feedback
- Adult learners refreshing math before a calculus-based college course
- Parents or teachers checking whether a student can explain ideas, not just produce answers
What you will be able to do
- Graph and transform polynomial, rational, exponential, log, and trig functions by hand
- Find all real and complex zeros of a polynomial and work confidently with complex numbers
- Solve exponential, logarithmic, and trigonometric equations, including over restricted intervals
- Prove trig identities and apply sum, difference, double, and half-angle formulas
- Solve systems of equations with substitution, elimination, and matrices, including 3x3 systems
- Work with vectors, polar coordinates, parametric equations, and the four conic sections
A taste of a lesson on Rational Functions and Asymptotes
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 28 topics in Precalculus
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Functions, Domain, and Range
A function assigns each input exactly one output; students learn function notation f(x), how to determine domain and range from equations, graphs, and real-world contexts, and how to test whether a relation is a function using the vertical line test.
Graphing and Transformations of Functions
How shifting, stretching, compressing, and reflecting a base function's graph (f(x)+k, f(x+h), af(x), f(-x), -f(x)) changes its equation and appearance, and how to combine multiple transformations in the correct order.
Builds on Functions, Domain, and Range
Combining and Composing Functions
Adding, subtracting, multiplying, and dividing functions; forming and evaluating composite functions f(g(x)); determining domains of combined and composite functions.
Builds on Functions, Domain, and Range
Inverse Functions
What it means for a function to be one-to-one, the horizontal line test, how to find an inverse algebraically, and the relationship between a function's graph and its inverse's graph (reflection across y=x).
Builds on Graphing and Transformations of Functions, Combining and Composing Functions
Linear and Quadratic Functions Review
Review and deepen understanding of linear functions (slope, intercepts, rate of change) and quadratic functions (vertex form, completing the square, the discriminant), setting the stage for polynomial functions.
Builds on Functions, Domain, and Range
Polynomial Functions and End Behavior
Degree, leading coefficient, and how they determine a polynomial's end behavior; identifying turning points and multiplicity of zeros from a graph; sketching polynomials from their factored form.
Builds on Graphing and Transformations of Functions, Linear and Quadratic Functions Review
Polynomial Division and Finding Zeros
Long division and synthetic division of polynomials, the Remainder and Factor Theorems, the Rational Root Theorem, and using these tools to find all real and complex zeros of a polynomial.
Builds on Polynomial Functions and End Behavior
Complex Numbers
Defining i as the square root of -1, performing arithmetic with complex numbers (addition, multiplication, conjugates, division), and understanding complex zeros of polynomials including the Fundamental Theorem of Algebra.
Builds on Polynomial Division and Finding Zeros
Rational Functions and Asymptotes
Graphing rational functions by identifying vertical, horizontal, and slant asymptotes and holes; understanding how numerator/denominator degree relationships determine end behavior and discontinuities.
Builds on Polynomial Division and Finding Zeros
Exponential Functions
Properties and graphs of exponential functions of the form f(x)=a·b^x, exponential growth and decay, the number e, and applications like compound interest and population models.
Builds on Graphing and Transformations of Functions
Logarithmic Functions
Logarithms as inverses of exponential functions, converting between exponential and logarithmic form, graphing log functions, common and natural logarithms, and properties/laws of logarithms used to expand, condense, and solve equations.
Builds on Inverse Functions, Exponential Functions
Solving Exponential and Logarithmic Equations
Techniques for solving equations involving exponential and logarithmic expressions, including using logarithm laws, checking for extraneous solutions, and applying these to real-world growth/decay problems.
Builds on Logarithmic Functions
Angles, Radians, and the Unit Circle
Measuring angles in degrees and radians and converting between them, coterminal and reference angles, and defining sine and cosine as coordinates of points on the unit circle.
Builds on Functions, Domain, and Range
Right Triangle Trigonometry
Defining sine, cosine, and tangent (and their reciprocals) as ratios of sides in a right triangle, solving for missing sides and angles, and connecting this definition to the unit circle definition.
Builds on Angles, Radians, and the Unit Circle
Graphing Trigonometric Functions
Graphing sine, cosine, and tangent functions and their transformations (amplitude, period, phase shift, vertical shift), and reading key features like maxima, minima, and zeros from these graphs.
Builds on Graphing and Transformations of Functions, Right Triangle Trigonometry
Inverse Trigonometric Functions
Restricting domains of trig functions to define arcsine, arccosine, and arctangent; evaluating inverse trig expressions and understanding their restricted ranges.
Builds on Inverse Functions, Graphing Trigonometric Functions
Fundamental Trigonometric Identities
Pythagorean, reciprocal, and quotient identities; using algebraic manipulation to simplify trig expressions and prove identities are equivalent.
Builds on Graphing Trigonometric Functions
Sum, Difference, Double, and Half-Angle Identities
Deriving and applying formulas for sin/cos/tan of sums and differences of angles, and double- and half-angle formulas, to simplify expressions and find exact values.
Builds on Fundamental Trigonometric Identities
Solving Trigonometric Equations
Solving equations involving trig functions over specified intervals or in general form, using identities, factoring, and inverse trig functions, and accounting for periodicity.
Builds on Inverse Trigonometric Functions, Sum, Difference, Double, and Half-Angle Identities
Law of Sines and Law of Cosines
Solving non-right (oblique) triangles using the Law of Sines and Law of Cosines, including the ambiguous case (SSA), and applying these laws to real-world problems.
Builds on Right Triangle Trigonometry, Solving Trigonometric Equations
Polar Coordinates and Graphs
Plotting points using polar coordinates, converting between polar and rectangular coordinates, and graphing simple polar equations such as circles, roses, and cardioids.
Builds on Angles, Radians, and the Unit Circle, Graphing Trigonometric Functions
Vectors in the Plane
Vectors as magnitude and direction, vector addition and scalar multiplication, component form, unit vectors, and applications such as combining forces or velocities.
Builds on Right Triangle Trigonometry
Systems of Equations and Inequalities
Solving systems of linear equations using substitution, elimination, and matrices (including row reduction for 2x2 and 3x3 systems), and solving/graphing systems of inequalities including linear programming basics.
Builds on Linear and Quadratic Functions Review
Matrices and Determinants
Matrix operations (addition, scalar multiplication, multiplication), finding determinants and inverses of matrices, and using matrices to solve systems of linear equations.
Builds on Systems of Equations and Inequalities
Sequences and Series
Arithmetic and geometric sequences and their explicit/recursive formulas, summation notation, and formulas for partial sums of arithmetic and geometric series including infinite geometric series.
Builds on Functions, Domain, and Range
The Binomial Theorem
Expanding binomial expressions using the Binomial Theorem and Pascal's Triangle, and using combinations to find specific terms in an expansion.
Builds on Sequences and Series
Conic Sections
Deriving and graphing the equations of circles, parabolas, ellipses, and hyperbolas from their geometric definitions (in terms of foci and directrices), and identifying key features (center, vertices, foci, asymptotes) from an equation.
Builds on Linear and Quadratic Functions Review, Systems of Equations and Inequalities
Parametric Equations
Defining curves using a parameter, converting between parametric and rectangular equations, and graphing parametric curves such as projectile motion paths.
Builds on Polar Coordinates and Graphs
Practice questions in Precalculus
A few of the kinds of questions you will work through, with answers.
If g(x) = -f(x - 3) + 2, describe how the graph of f is transformed to get g.
See the answer
Shift right 3, reflect over the x-axis, shift up 2.
The -3 inside shifts right, the negative sign outside reflects vertically, and +2 shifts the whole graph up.
Use synthetic division to divide x³ - 4x² + x + 6 by (x - 2), then state the quotient.
See the answer
x² - 2x - 3
Synthetic division with root 2 gives coefficients 1, -2, -3 and remainder 0, so (x - 2) is a factor.
Solve for x: log₂(x) + log₂(x - 2) = 3
See the answer
x = 4
Combine logs to log₂[x(x-2)] = 3, so x² - 2x = 8, giving x = 4 or x = -2; reject x = -2 since log of a negative number is undefined.
Simplify: (1 - cos²θ) / sinθ
See the answer
sinθ
Since 1 - cos²θ = sin²θ, the expression becomes sin²θ / sinθ = sinθ.
A triangle has sides a = 7, b = 10, and included angle C = 40°. Find side c.
See the answer
c ≈ 6.5
Using the Law of Cosines, c² = 7² + 10² - 2(7)(10)cos(40°) ≈ 42.4, so c ≈ 6.5.
Identify the conic and find its center: (x - 1)²/9 + (y + 2)²/4 = 1
See the answer
Ellipse centered at (1, -2)
The equation matches the standard ellipse form with a² = 9 and b² = 4, centered at (h, k) = (1, -2).
Questions about this subject
Do I need to know trigonometry before starting?
No. The course builds trigonometry from scratch starting with angles, radians, and the unit circle, assuming only solid Algebra I and II skills.
Is this enough preparation for calculus?
The course covers the 28 topics typically expected before a calculus course, including polynomial, rational, exponential, logarithmic, and trigonometric functions, but it does not include limits, derivatives, or integrals.
How are sequences, series, and the binomial theorem covered?
These come after the core function and trigonometry work, covering arithmetic and geometric sequences, summation notation, partial and infinite sums, and binomial expansion with Pascal's Triangle.
What if I'm weak on an earlier topic, like factoring or function notation?
The tutor checks your answers as you go and can step back to Algebra II-level review on the spot if a gap shows up, such as revisiting factoring before tackling polynomial zeros.
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