Subjects › Algebra 2
Algebra 2: functions, polynomials, trig, and statistics by talking it through
This page covers a full year of Algebra 2 on the Common Core traditional pathway: sequences and functions, transformations and inverses, polynomial, rational, radical, exponential, logarithmic, and trigonometric functions, complex numbers, systems and matrices, and statistical inference. It picks up right where Algebra 1 and Geometry basics leave off and stops before calculus and formal probability theory.
A lesson starts from something you can already do, like factoring a quadratic or graphing a line, and the tutor asks a question that pushes one step further, like why that quadratic's graph touches the x-axis only once. You answer in your own words or by working out a step, the tutor responds to what you actually wrote, and the dialogue moves forward only when you can both explain the idea and carry out the procedure. Mistakes get used as the next question, not corrected and skipped past.
Who it is for
- Students currently enrolled in Algebra 2 who want to work through topics in order or get unstuck on one
- Students who finished Algebra 1 and Geometry and want a head start before the course starts
- Parents who want to see what their student is actually being asked to explain, not just the answer
- Teachers looking for a second channel of practice that asks for reasoning, not just final answers
- Students retaking or reviewing Algebra 2 before a placement test or a later course that assumes it
What you will be able to do
- Factor, solve, and graph polynomial, rational, radical, exponential, logarithmic, and trigonometric functions, and explain what each graph's features mean
- Work with complex numbers and explain why a degree-n polynomial has n complex zeros
- Build a function model from a table, graph, or description, including exponential growth and periodic phenomena
- Solve systems of linear equations, linear-quadratic systems, and matrix equations, by hand and with matrices
- Use the unit circle and trig identities to find exact values and solve trig equations
- Read a statistical study, judge whether a sample size or difference is meaningful, and explain correlation versus causation
A taste of a lesson on Polynomial zeros and rough graphs
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 48 topics in Algebra 2
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Arithmetic and geometric sequences
Sequences as functions on the integers. Arithmetic sequences grow by a constant difference; geometric sequences grow by a constant ratio. Write each recursively (a_(n+1) = a_n + d, a_(n+1) = r*a_n) and with an explicit formula (a_n = a_1 + (n-1)d, a_n = a_1 r^(n-1)), and translate between the two forms. Include the Fibonacci sequence as a recursive example and use sequences to model contexts such as savings or depreciation.
Functions, function notation, domain and range
A function assigns exactly one output to each input in its domain. Use f(x) notation to evaluate functions and interpret statements like f(3)=10 in context. Identify domain and range, including restrictions imposed by context (e.g., number of people must be a whole number) and by the expression (no division by zero, no negative numbers under even roots).
Linear and quadratic functions: graphs, intercepts, extremes
Graph linear and quadratic functions showing intercepts, slope, vertex, maximum and minimum, and axis of symmetry. Connect the vertex form and standard form to the graph. Interpret intercepts and extreme values in context.
Builds on Functions, function notation, domain and range
Linear versus exponential growth
Linear functions change by equal differences over equal intervals; exponential functions change by equal factors over equal intervals. Recognize from a description, table, or graph whether a situation is linear (constant rate per unit) or exponential (constant percent rate per unit). Prove why each pattern holds, and observe that exponentials eventually exceed linear and polynomial growth.
Builds on Arithmetic and geometric sequences, Linear and quadratic functions: graphs, intercepts, extremes
Constructing linear and exponential models
Build a linear or exponential function from a graph, a verbal description, two input-output pairs, or a table. Interpret the parameters (slope and intercept; initial value and growth or decay factor) in context, and identify percent rate of change from a base such as 1.02 or 0.97.
Builds on Linear versus exponential growth
Properties of exponents and rational exponents
Integer and rational exponents: x^a x^b = x^(a+b), (x^a)^b = x^(ab), x^0 = 1, negative exponents as reciprocals, and a^(1/n) as the nth root. Rewrite exponential expressions to reveal growth factors, e.g., 1.15^t = (1.15^(1/12))^(12t) ≈ 1.012^(12t) to get an equivalent monthly rate.
Builds on Constructing linear and exponential models
Polynomial arithmetic and closure
Polynomials are closed under addition, subtraction, and multiplication, just as integers are. Add, subtract, and multiply polynomials using the distributive property, combining like terms, and organizing by degree. Recognize that the degree of a product is the sum of the degrees.
Builds on Properties of exponents and rational exponents
Interpreting and restructuring expressions
Interpret the terms, factors, and coefficients of an expression in context. View complicated parts as a single entity (e.g., in P(1+r)^n, treat (1+r) as one factor). Use structure to rewrite expressions, such as seeing x^4 - y^4 as a difference of squares.
Builds on Polynomial arithmetic and closure
Factoring quadratics and polynomials
Factor quadratics (trinomials, difference of squares, perfect squares), and factor higher-degree polynomials by greatest common factor, grouping, sum and difference of cubes, and recognizing patterns. Use factored form to reveal zeros of the function.
Builds on Interpreting and restructuring expressions
Polynomial identities
Prove polynomial identities such as (a+b)^2 = a^2 + 2ab + b^2, (a-b)(a+b) = a^2 - b^2, and (x^2+y^2)^2 = (x^2-y^2)^2 + (2xy)^2 by expanding and comparing both sides. Use identities to describe numerical relationships, such as generating Pythagorean triples, and extend identities to complex numbers (e.g., x^2+4 = (x+2i)(x-2i)) later.
Builds on Factoring quadratics and polynomials
Binomial Theorem and Pascal's Triangle
Expand (x+y)^n for positive integer n using the Binomial Theorem, with coefficients from Pascal's Triangle or binomial coefficients nCk. Understand why each coefficient counts ways to choose k of n factors. Apply to find specific terms of an expansion.
Builds on Polynomial identities
Solving linear equations and inequalities with parameters
Solve linear equations and inequalities in one variable, including equations with letter coefficients (solve for a specified variable). Justify each step as following from equality (or, for inequalities, reversing sign when multiplying by a negative). Explain why a solution assumed to exist can be checked by reversing steps.
Builds on Linear and quadratic functions: graphs, intercepts, extremes
Solving quadratic equations by multiple methods
Solve quadratics by inspection, square roots (x^2 = 49), factoring, completing the square, and the quadratic formula; choose the method suited to the form. Derive the quadratic formula by completing the square on ax^2+bx+c=0 to get (x-p)^2 = q. Use the discriminant to recognize when solutions are real, repeated, or non-real.
Builds on Factoring quadratics and polynomials, Solving linear equations and inequalities with parameters
Completing the square to reveal vertex and extremes
Complete the square in a quadratic expression to rewrite it as a(x-h)^2 + k, revealing the vertex, maximum or minimum value, and axis of symmetry. Interpret the maximum or minimum in context (e.g., maximum revenue). Connect to the quadratic formula derivation.
Builds on Solving quadratic equations by multiple methods
Complex numbers: arithmetic and conjugates
Know i with i^2 = -1 and write every complex number as a+bi. Add, subtract, and multiply complex numbers using the distributive, commutative, and associative properties with i^2 = -1. Find complex conjugates and use them to divide complex numbers and compute moduli.
Builds on Solving quadratic equations by multiple methods
Complex solutions and the Fundamental Theorem of Algebra
Solve quadratics with real coefficients that have non-real complex solutions via the quadratic formula. State the Fundamental Theorem of Algebra (a degree-n polynomial has n complex zeros counting multiplicity) and show it for quadratics. Connect non-real roots to complex conjugate pairs for real-coefficient polynomials.
Builds on Polynomial identities, Complex numbers: arithmetic and conjugates
Remainder Theorem and the Factor Theorem
For a polynomial p(x) and a number a, the remainder on division by (x - a) equals p(a). Therefore p(a)=0 exactly when (x-a) is a factor. Carry out polynomial long division and synthetic division, and use these to test candidate zeros.
Builds on Polynomial arithmetic and closure, Factoring quadratics and polynomials
Polynomial zeros and rough graphs
Identify zeros of a polynomial from a suitable factored form, noting multiplicity (crossing versus touching at zeros). Use zeros, end behavior, and y-intercept to sketch a rough graph of the polynomial function. Graph polynomial functions showing end behavior.
Builds on Remainder Theorem and the Factor Theorem
Rational expressions: simplify, add, subtract, multiply, divide
Rational expressions are closed under addition, subtraction, multiplication, and division by a nonzero rational expression, analogous to rational numbers. Simplify by factoring and cancelling, state excluded values, and perform all four operations, finding common denominators for sums and differences.
Builds on Remainder Theorem and the Factor Theorem
Rewriting rational expressions as quotient plus remainder
Write a(x)/b(x) in the form q(x) + r(x)/b(x) where deg r < deg b, using polynomial long division. Explain why this form helps analyze the expression's behavior for large inputs and the associated asymptote.
Builds on Rational expressions: simplify, add, subtract, multiply, divide
Graphing rational functions
Graph rational functions identifying zeros (from numerator), vertical asymptotes (from denominator factors not cancelled), holes from common factors, horizontal or slant asymptotes from end behavior, and intercepts. Use the quotient-plus-remainder form to show end behavior.
Builds on Polynomial zeros and rough graphs, Rewriting rational expressions as quotient plus remainder
Solving rational equations and recognizing extraneous solutions
Solve rational equations by multiplying through by the least common denominator, then check candidates against excluded values. Give examples showing how extraneous solutions arise from squaring or multiplying by expressions that can be zero.
Builds on Solving quadratic equations by multiple methods, Rational expressions: simplify, add, subtract, multiply, divide
Radical expressions and radical equations
Simplify radicals and rational exponents, rationalize denominators, and solve radical equations by isolating the radical and raising both sides to a power. Check for extraneous solutions introduced by squaring. Graph square root and cube root functions and identify their domains.
Builds on Properties of exponents and rational exponents, Solving quadratic equations by multiple methods
Transformations of function graphs
Describe the effect on a graph of replacing f(x) with f(x)+k, kf(x), f(kx), and f(x+k) for positive and negative k. Given a graph, find k; use transformations to graph functions such as absolute value, square root, cube root, and piecewise and step functions. Explain the transformations algebraically.
Builds on Functions, function notation, domain and range, Linear and quadratic functions: graphs, intercepts, extremes
Combining and composing functions
Combine functions by addition, subtraction, multiplication, and division, and compose functions f(g(x)). Build a model from standard function types, such as adding a constant to a decaying exponential to model a cooling body, and relate the composite to the model. Compose functions in context, e.g., T(h(t)).
Builds on Constructing linear and exponential models, Transformations of function graphs
Inverse functions
Find the inverse of a function with an inverse by solving f(x)=c for x and swapping variables, and verify inverses by checking f(f^-1(x))=x and f^-1(f(x))=x. Read inverse values from a graph or table. Produce an invertible function by restricting the domain of a non-invertible one (e.g., x^2 on x≥0). Note that inverse graphs reflect over y=x.
Builds on Radical expressions and radical equations, Combining and composing functions
Exponential and logarithmic functions and their graphs
Understand logarithms as inverses of exponentials: log_b(x)=y iff b^y=x. Graph exponential and logarithmic functions showing intercepts, asymptotes, and end behavior. Use the inverse relationship to solve exponential and logarithmic equations, and evaluate logs using technology, including base 10 and base e.
Builds on Properties of exponents and rational exponents, Inverse functions
Logarithm properties and exponential models
Apply log(MN)=log M + log N, log(M/N), log(M^p)=p log M, and change of base. Solve ab^(ct)=d by expressing the solution as a logarithm and evaluating with technology. Use exponential models in context, e.g., doubling time and half-life, and interpret parameters.
Builds on Exponential and logarithmic functions and their graphs
Sums of finite geometric series
Derive the formula for the sum of a finite geometric series with common ratio not equal to 1, S_n = a_1(1-r^n)/(1-r), by multiplying by r and subtracting. Use the formula to solve problems such as mortgage payments and savings plans.
Builds on Arithmetic and geometric sequences, Properties of exponents and rational exponents
Radian measure and the unit circle
Define radian measure as arc length on the unit circle (angle in radians = arc length for radius 1). Extend sine and cosine to all real numbers via the unit circle traversed counterclockwise, with sin = y-coordinate and cos = x-coordinate. Convert between degrees and radians.
Builds on Linear and quadratic functions: graphs, intercepts, extremes, Properties of exponents and rational exponents
Complex plane, polar form, and geometry of complex operations
Plot complex numbers on the complex plane, including real and imaginary numbers. Represent a+bi in rectangular and polar form (r, theta), explain why both describe the same number, and represent addition, subtraction, multiplication, and conjugation geometrically. Compute distance between numbers as the modulus of the difference and midpoints as averages; use polar form to check results such as (-1+sqrt(3)i)^3 = 8.
Builds on Complex numbers: arithmetic and conjugates, Radian measure and the unit circle
Special-angle values, symmetry, and periodicity
Use special right triangles (30-60-90 and 45-45-90) to find sine, cosine, and tangent at pi/6, pi/4, pi/3, and use the unit circle to find values at pi-x, pi+x, and 2pi-x. Explain odd and even symmetry of sine, cosine, and tangent, and periodicity.
Builds on Radian measure and the unit circle
Graphing trig functions and modeling periodic phenomena
Graph y=A sin(Bx+C)+D and cosine; identify amplitude, period, midline, and phase shift. Choose a trigonometric function to model periodic data (e.g., tides, daylight) given amplitude, frequency, and midline. Connect transformations to parameters.
Builds on Transformations of function graphs, Special-angle values, symmetry, and periodicity
Pythagorean identity and the other trig functions
Prove sin^2(θ)+cos^2(θ)=1 from the unit circle and use it to find sin, cos, or tan given one value and the quadrant. Define secant, cosecant, cotangent and tangent as sin/cos, and relate them via identities.
Builds on Special-angle values, symmetry, and periodicity
Inverse trig functions and solving trig equations
Restrict sine, cosine, and tangent to intervals where they are one-to-one to define inverses (arcsin, arccos, arctan). Use inverses to solve trigonometric equations arising in modeling contexts, evaluate solutions with technology, and interpret them in context, including all solutions in a period.
Builds on Inverse functions, Graphing trig functions and modeling periodic phenomena, Pythagorean identity and the other trig functions
Addition and subtraction formulas for trigonometry
Prove the sine, cosine, and tangent addition and subtraction formulas (e.g., from geometric or unit-circle arguments) and use them to evaluate exact trig values such as sin(75°) and to simplify expressions.
Builds on Pythagorean identity and the other trig functions
Systems of linear equations in two variables
Solve pairs of linear equations exactly by substitution and elimination, and approximately with graphs. Explain why replacing one equation by the sum of itself and a multiple of the other preserves the solution set. Interpret solutions in context, including no solution and infinitely many solutions.
Builds on Linear and quadratic functions: graphs, intercepts, extremes, Solving linear equations and inequalities with parameters
Systems of linear inequalities
Graph a linear inequality in two variables as a half-plane, using a dashed boundary for strict inequality and solid for non-strict, and test a point. Graph a system of linear inequalities as the intersection of half-planes, identifying the feasible region.
Builds on Systems of linear equations in two variables
Graphs of equations and intersections of functions
The graph of an equation in two variables is the set of its solutions in the coordinate plane. The x-coordinates where y=f(x) and y=g(x) intersect are the solutions of f(x)=g(x); find them graphically and with technology, approximately when needed.
Builds on Transformations of function graphs, Systems of linear equations in two variables
Systems of a linear and a quadratic equation
Solve a linear equation with a quadratic equation (e.g., line y=-3x and circle x^2+y^2=3) algebraically by substitution and reducing to a quadratic, and graphically by finding intersection points. Interpret the number of intersections (0, 1, or 2) and relate to discriminant.
Builds on Solving quadratic equations by multiple methods, Graphs of equations and intersections of functions
Matrices: representing and solving linear systems
Represent a system of linear equations as a single matrix equation AX=B in a vector variable. Find the inverse of a 2x2 matrix (and larger matrices with technology), and use it to solve the system X = A^(-1)B when the inverse exists.
Builds on Systems of linear equations in two variables
Displaying and comparing data distributions
Represent one-variable data with dot plots, histograms, and box plots on the real number line. Compare center (median, mean) and spread (IQR, standard deviation) appropriately to the shape of each distribution, and interpret differences in shape, center, and spread in context, accounting for outliers. Fit a normal distribution with mean and standard deviation to estimate percentages, and recognize when this is inappropriate.
Builds on Functions, function notation, domain and range
Two-way frequency tables and association
Summarize categorical data for two categories in two-way tables. Compute joint, marginal, and conditional relative frequencies, interpret them in context, and recognize possible associations and trends.
Builds on Displaying and comparing data distributions
Scatter plots, linear and function models, residuals
Represent bivariate data on a scatter plot and describe the relationship. Fit a linear function to data showing linear association, and fit other functions (exponential, quadratic) to data as appropriate. Interpret slope and intercept of a linear model in context, and assess fit by plotting and analyzing residuals.
Builds on Constructing linear and exponential models, Displaying and comparing data distributions
Correlation coefficient and correlation versus causation
Compute and interpret the correlation coefficient r of a linear fit using technology, understanding its sign and strength. Distinguish correlation from causation, giving examples of confounding and lurking variables.
Builds on Scatter plots, linear and function models, residuals
Statistics as inference: surveys, experiments, observational studies
Statistics makes inferences about population parameters from random samples. Recognize the purposes and differences among sample surveys, experiments, and observational studies, and explain how randomization supports causal conclusions in experiments and representative estimates in surveys.
Builds on Displaying and comparing data distributions
Simulation models and sampling variability
Decide whether a specified probability model is consistent with observed results by simulating the data-generating process (e.g., would 5 tails in a row cast doubt on a fair coin?). Use simulation to build margins of error for sample proportions and means, and to judge whether differences between two treatments are significant.
Builds on Statistics as inference: surveys, experiments, observational studies
Estimating population parameters and comparing treatments
Use sample survey data to estimate a population mean or proportion with a margin of error. Use data from a randomized experiment to compare two treatments, judging whether observed differences are larger than chance variation via simulation. Evaluate reports based on data, checking sampling method, sample size, and whether conclusions overreach.
Builds on Simulation models and sampling variability
Practice questions in Algebra 2
A few of the kinds of questions you will work through, with answers.
Simplify: 8^(2/3)
See the answer
4
8^(1/3) = 2 (cube root of 8), then square it: 2^2 = 4.
Factor x^2 - 5x - 14
See the answer
(x - 7)(x + 2)
Need two numbers multiplying to -14 and adding to -5: -7 and 2 work.
Multiply (3 + 2i)(1 - i)
See the answer
5 - i
Expand: 3 - 3i + 2i - 2i^2 = 3 - i + 2 = 5 - i, using i^2 = -1.
For f(x) = (x^2 - 9)/(x - 3), what feature appears at x = 3: a hole or a vertical asymptote?
See the answer
A hole
x^2 - 9 factors as (x-3)(x+3), so the (x-3) cancels with the denominator, leaving a removable discontinuity rather than an asymptote.
Solve for x: log_2(x) = 5
See the answer
x = 32
log_2(x) = 5 means 2^5 = x, and 2^5 = 32.
Find the exact value of sin(75°) using sin(45° + 30°)
See the answer
(√6 + √2)/4
sin(45+30) = sin45 cos30 + cos45 sin30 = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
Questions about this subject
Do I need trigonometry or precalculus already?
No. Algebra 2 introduces trig here for the first time, starting from radian measure and the unit circle, assuming only the right-triangle and Pythagorean theorem basics from Geometry.
Does this cover the statistics part of Algebra 2, or just the algebra?
Both. The course includes data distributions, two-way tables, correlation, and statistical inference from sampling and experiments, alongside the function and equation topics.
How does the tutor handle a topic I'm shaky on from Algebra 1, like factoring?
If a question reveals a gap, the tutor steps back to ask about the underlying Algebra 1 skill before continuing, rather than assuming it's solid.
Is this enough to prepare for Precalculus?
It covers the Common Core Algebra 2 scope in full, including functions, polynomials, logs, trig, and complex numbers, which is what most Precalculus courses assume; it stops before calculus-level content.