Looking for free content that’s aligned to your standards? You’ve come to the right place!

## Get Free Precalculus Math Content

Khan Academy is a nonprofit with thousands of free videos, articles, and practice questions for just about every standard.

No ads, no subscriptions – just 100% free, forever.

### N Number and Quantity

- N.VM Vector and Matrix Quantities
- Represent and model with vector quantities.
- N.VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
- N.VM.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- N.VM.3 Solve problems involving velocity and other quantities that can be represented by vectors.

- Perform operations on vectors.
- N.VM. 4 Add and subtract vectors.
- N.VM. 4.a Add vectors end to end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
- N.VM. 4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
- N.VM. 4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

- N.VM.5 Multiply a vector by a scalar.
- N.VM.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx , vy) = (cvx, cvy).
- N.VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against vs (for c < 0).

- N.VM. 4 Add and subtract vectors.
- Perform operations on matrices and use matrices in applications.
- N.VM.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- N.VM.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- N.VM.8 Add, subtract, and multiply matrices of appropriate dimensions.
- N.VM.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- N.VM.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- N.VM.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- N.VM.12 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
- N.VM.13 Solve systems of linear equations up to three variables using matrix row reduction.

- Represent and model with vector quantities.
- N.CN Complex Number Systems
- Perform arithmetic operations with complex numbers.
- N.CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

- Represent complex numbers and their operations on the complex plane.
- N.CN.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- N.CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)³ = 8, because (-1 + √3 i) has modulus 2 and argument 120°.
- N.CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

- Use complex numbers in polynomial identities and equations.
- N.CN.10 Multiply complex numbers in polar form and use DeMoivre’s Theorem to find roots of complex numbers.

- Perform arithmetic operations with complex numbers.

### A Algebra

- A.REI Reasoning with Equations and Inequalities
- Solve systems of equations.
- A.REI.8. Represent a system of linear equations as a single matrix equation in a vector variable.
- A.REI.9. Find the inverse of a matrix, if it exists, and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).

- Solve systems of equations.

### F Functions

- F.IF Interpreting Functions
- Analyze functions using different representations.
- F.IF.7 Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
- F.IF.7.d Graph rational functions, identifying zeros, asymptotes, and point discontinuities when suitable factorizations are available, and showing end behavior.
- F.IF.7.f Define a curve parametrically and draw its graph.

- F.IF.10 Use sigma notation to represent the sum of a finite arithmetic or geometric series.
- F.IF.11 Represent series algebraically, graphically, and numerically.

- F.IF.7 Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

- Analyze functions using different representations.
- F.BF Building Functions
- Build a function that models a relationship between two quantities.
- F.BF.1 Write a function that describes a relationship between two quantities.
- F.BF.1.c Compose functions.

- F.BF.1 Write a function that describes a relationship between two quantities.
- Build new functions from existing functions.
- F.BF.4 Find inverse functions.
- F.BF.4.b Verify by composition that one function is the inverse of another.
- F.BF.4.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
- F.BF.4.d Produce an invertible function from a non-invertible function by restricting the domain.

- F.BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

- F.BF.4 Find inverse functions.

- Build a function that models a relationship between two quantities.
- F.TF Trigonometric Functions
- Extend the domain of trigonometric functions using the unit circle.
- F.TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

- Model periodic phenomena with trigonometric functions.
- F.TF.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
- F.TF.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context

- Prove and apply trigonometric identities.
- F.TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

- Extend the domain of trigonometric functions using the unit circle.

### G Geometry

- G.GMD Geometric Measurement and Dimension
- Explain volume formulas and use them to solve problems.
- G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

- Explain volume formulas and use them to solve problems.
- G.GPE Expressing Geometric Properties With Equations
- Translate between the geometric description and the equation for a conic section.
- G.GPE.2 Derive the equation of a parabola given a focus and a directrix.
- G.GPE.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

- Translate between the geometric description and the equation for a conic section.

### S Statistics

- S.CP Conditional Probability and the Rules of Probability
- Understand independence and conditional probability and use them to interpret data.
- S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
- S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of B given A is the same as the probability of B.

- Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
- S.CP.8 Apply the general Multiplication Rule in a uniform probability model, P(A andB) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
- S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

- Understand independence and conditional probability and use them to interpret data.