SAT · SAT Math domains
Passport to Advanced Math
Exponent rules, radicals and expressions with rational exponents.
Radicals: evaluating and simplifying square and cube roots, solving x^2 = p.
Radicals: simplifying, multiplying and rationalising square-root expressions.
Rational equations and expressions, including extraneous solutions.
A note on this mapping: This skill straddles two domains: clearing denominators in a linear equation is Heart of Algebra, while a genuine rational equation is Passport to Advanced Math.
Factoring quadratics and using the structure of an expression.
Solving quadratic equations and interpreting their solutions.
Adding, subtracting and multiplying polynomials, including the special products.
Isolating a quantity of interest in a formula (literal equations).
Function notation, evaluation, domain and range.
Exponential expressions, functions and their graphs.
Interpreting and manipulating non-linear functions, including shifts of a graph.
A note on this mapping: Transformations are tested through the questions rather than named as a topic; this is the domain those questions are counted in.
Vertex, axis of symmetry, zeros and the forms of a quadratic function.
Polynomial division, the remainder theorem, zeros and the shape of a polynomial graph.
Simplifying, combining and solving with rational expressions.
Rational exponents, radicals and equations with a variable under a root.
The domain, asymptotes and graph of a rational function.
Exponential expressions and the relationship between exponents and logarithms.
A note on this mapping: The SAT tests exponential structure and almost never a logarithm by name, so this skill goes further than the test does.
Solving exponential equations, including those that need a logarithm.
Arithmetic and geometric sequences as linear and exponential functions.
Sums of arithmetic and geometric sequences.
A note on this mapping: Series appear rarely on the SAT, and then as a pattern to reason about rather than a formula to apply.
Evaluating and simplifying a composition of two functions.
Finding and interpreting the inverse of a function, and reading one off a table.
The graph of a transformed parent function and the effect of each constant.
Building a function from a described situation and interpreting its parameters.
The 24 skills that cover it
- ExponentsUse exponents as shorthand for repeated multiplication, and apply the product, quotient, zero and negative exponent rules.Read the lesson →
- Square & Cube RootsUndo a square or a cube: find the root of a perfect square or cube, solve x^2 = p and x^3 = p, and estimate the roots that are not whole numbers.Read the lesson →
- Simplifying RadicalsWrite a square root in simplest form by pulling out perfect-square factors, then add, multiply and divide radicals, and clear a root from a denominator.Read the lesson →
- Equations with FractionsClear fractions from an equation by multiplying through by the least common denominator, then solve as usual.Read the lesson →
- FactoringRewrite a polynomial as a product: pull out the greatest common factor, factor trinomials, and recognise the difference of squares.Read the lesson →
- Quadratic EquationsSolve equations containing a squared term by factoring, by taking square roots, and with the quadratic formula.Read the lesson →
- PolynomialsName a polynomial by its degree and number of terms, add and subtract polynomials, multiply a monomial or a binomial through, and recognise the special products.Read the lesson →
- Literal Equations & FormulasRearrange a formula to isolate any one of its letters, using the same moves as a numeric equation, and use the rearranged form to solve for the quantity you actually need.Read the lesson →
- Function NotationRead f(x) as a machine that turns an input into exactly one output, and evaluate and interpret function notation.Read the lesson →
- Exponential FunctionsModel repeated growth and decay with y = a * b^x, and contrast constant differences with constant ratios.Read the lesson →
- Function TransformationsShift, stretch and reflect a parent graph, and read the transformation directly off the equation.Read the lesson →
- Quadratic Functions & ParabolasRead a parabola from its equation: find the vertex, the axis of symmetry and the zeros of y = ax^2 + bx + c, convert to vertex form by completing the square, and use the vertex to answer maximum and minimum questions.Read the lesson →
- Polynomial FunctionsDivide one polynomial by another, use the remainder and factor theorems to find zeros, and read the end behaviour and shape of a polynomial graph from its degree and leading coefficient.Read the lesson →
- Rational Expressions & EquationsTreat a fraction of polynomials the way you treat a fraction of numbers: simplify it by factoring, state the values it excludes, multiply, divide, add and subtract, and solve equations that contain one.Read the lesson →
- Rational Exponents & Radical EquationsRead a fractional exponent as a root, move between radical and exponent form, simplify with the exponent rules, and solve equations with a variable under a root, checking for solutions the squaring step invents.Read the lesson →
- Rational FunctionsFind where a rational function is undefined and what its graph does there: vertical and horizontal asymptotes, holes, intercepts and end behaviour, and use them to sketch and read the graph.Read the lesson →
- LogarithmsRead a logarithm as the exponent that answers "b to what power gives x", convert between exponential and logarithmic form, evaluate exact logs, and expand or condense with the product, quotient and power laws.Read the lesson →
- Exponential & Logarithmic EquationsSolve for an exponent by taking a logarithm and for a logarithm by rewriting in exponential form, check every answer against the domain, and use the method on doubling times, pH, decibels and radioactive dating.Read the lesson →
- Arithmetic & Geometric SequencesSee a list of numbers as a function of its position: tell an arithmetic sequence from a geometric one, write explicit and recursive formulas, and find any term without listing the ones before it.Read the lesson →
- Series & Sigma NotationAdd the terms of a sequence without adding them one by one: the arithmetic series formula, the finite geometric series formula, sigma notation, and the infinite geometric series that settles on a value.Read the lesson →
- Composite FunctionsFeed one function into another: evaluate (f ∘ g)(a) from rules and tables, write f(g(x)) as one expression, find the domain of a composition, and take a composite function apart again.Read the lesson →
- Inverse FunctionsRun a function backwards: find the inverse of a linear, rational, radical, cubic, exponential or logarithmic function, verify two functions are inverses by composing them, read an inverse off a table or a graph, and decide when an inverse is a function at all.Read the lesson →
- Transformations of Any FunctionMove past the parabola: stretch a graph horizontally with f(kx), reflect it in the y-axis, apply several transformations in the right order, track where a single point goes, and do it all to square roots, cube roots, reciprocals, cubics and exponentials.Read the lesson →
- Piecewise Functions & ModellingBuild the function a situation needs: evaluate and write piecewise and step functions, find the average rate of change on an interval, turn a box, a fence or a price into a polynomial model, choose the right family, and use direct, inverse and joint variation.Read the lesson →
What a student should be able to do
- Evaluate a power such as 34 and explain what the exponent counts.
- Apply the product and quotient rules for powers with the same base.
- Interpret a zero exponent and a negative exponent.
- Evaluate powers inside expressions with the correct order of operations.
- Evaluate square roots of perfect squares and cube roots of perfect cubes.
- Solve equations of the form x2 = p and x3 = p, remembering both square roots.
- Estimate a non-perfect square root between two whole numbers and to one decimal place.
- Use roots to find the side of a square or the edge of a cube from its area or volume.
- Simplify a square root such as sqrt(50) to 5 sqrt(2) by factoring out the largest perfect square.
- Multiply and divide square roots, and simplify the product.
- Add and subtract like radicals after simplifying each one.
- Rationalise a denominator such as 6 / sqrt(3).
- Multiply both sides by the least common denominator to clear fractions.
- Solve equations with fractional coefficients and constants.
- Solve simple proportions as rational equations.
- Check solutions and exclude values that make a denominator zero.
- Factor out the greatest common factor of a polynomial.
- Factor a trinomial of the form x2 + bx + c.
- Recognise and factor a difference of two squares.
- Multiply the factors back to check the result.
- Solve a quadratic by factoring and the zero-product property.
- Solve x2 = k by taking square roots, keeping both signs.
- Apply the quadratic formula and interpret the discriminant.
- Solve projectile and area word problems with quadratics.
- Write a polynomial in standard form and state its degree and its number of terms.
- Add and subtract polynomials by combining like terms.
- Multiply a monomial by a polynomial and a binomial by a binomial.
- Expand the special products (a + b)2, (a - b)2 and (a + b)(a - b).
- Solve a formula such as p = 2l + 2w for one of its variables.
- Isolate a variable that appears inside a fraction or multiplied by another letter.
- Choose the rearrangement that matches a given formula.
- Rearrange a formula and then evaluate it for given values.
- Evaluate f(x) for a given input.
- Solve f(x) = k for the input that gives a required output.
- Decide whether a relation is a function.
- State the domain and range of a simple function.
- Evaluate an exponential function at a given input.
- Identify growth and decay from the base.
- Write an exponential model from a starting amount and a rate.
- Compare linear and exponential growth over time.
- Describe the effect of f(x) + k and f(x + h).
- Describe vertical stretches, compressions and reflections.
- Write the equation of a transformed parent function.
- Sketch a transformed graph from its equation.
- Find the vertex and the axis of symmetry of y = ax2 + bx + c.
- Find the zeros of a quadratic function and say where its graph crosses the x-axis.
- Complete the square to write a quadratic in vertex form, and read the vertex from it.
- Use the vertex to find the greatest height or the largest area in an application.
- Divide a polynomial by a linear factor using long or synthetic division, and write the quotient and remainder.
- Use the remainder theorem to evaluate p(k) and the factor theorem to decide whether x - k is a factor.
- Find all the zeros of a factorable polynomial, state each multiplicity, and say whether the graph crosses or touches at each.
- Describe the end behaviour of a polynomial from its degree and leading coefficient, and match a polynomial to its graph.
- Simplify a rational expression by factoring, and state the values of the variable that are excluded.
- Multiply and divide rational expressions, cancelling common factors.
- Add and subtract rational expressions using a least common denominator.
- Solve a rational equation by clearing denominators, and reject any solution that makes a denominator zero.
- Evaluate an expression with a rational exponent such as 82/3, and take an nth root.
- Convert between radical form and rational-exponent form, and simplify using the exponent rules.
- Solve a radical equation by isolating the root and raising both sides to a power.
- Check every candidate in the original equation and reject extraneous solutions; state the domain of a radical function.
- State the domain of a rational function and tell a vertical asymptote from a hole.
- Find the horizontal asymptote from the degrees of the numerator and denominator.
- Find the intercepts and the coordinates of a hole.
- Match a rational function to its graph, and describe its behaviour near an asymptote.
- Convert an equation between exponential form and logarithmic form.
- Evaluate a logarithm exactly, including common and natural logarithms.
- Expand and condense logarithmic expressions with the product, quotient and power laws.
- Use the change-of-base formula and estimate a logarithm from a calculator.
- Solve an exponential equation by matching bases or by taking a logarithm of both sides.
- Solve a logarithmic equation by rewriting it in exponential form or by condensing first.
- Reject a solution that makes the argument of a logarithm zero or negative.
- Find the time for an exponential model to reach a given amount.
- Identify a sequence as arithmetic or geometric and find its common difference or ratio.
- Write an explicit formula for the nth term and use it to find a given term.
- Write and use a recursive formula, and convert between recursive and explicit forms.
- Find a formula from two given terms.
- Find the sum of the first n terms of an arithmetic series.
- Find the sum of the first n terms of a geometric series.
- Read and write a series in sigma notation and evaluate it.
- Decide whether an infinite geometric series converges and find its sum.
- Evaluate (f ∘ g)(a) and (g ∘ f)(a) from function rules and from tables, and explain why the order matters.
- Write f(g(x)) as a single simplified expression when the outer function is not linear.
- State the domain of a composition by tracking the restrictions of both functions.
- Decompose a function into an outer and an inner function, and model a two-stage situation with a composition.
- Find the inverse of a function algebraically by swapping x and y and solving.
- Verify that two functions are inverses by showing f(g(x)) = x and g(f(x)) = x.
- Read values of an inverse from a table or a graph, and reflect a graph in the line y = x.
- Use the horizontal line test, and restrict a domain so that a function has an inverse.
- Describe the effect of f(kx) and f(-x), and combine horizontal and vertical transformations in the correct order.
- Find the image of a point on y = f(x) under y = a f(b(x - h)) + k.
- Write the equation of a transformed square root, cube root, reciprocal, cubic, absolute value or exponential parent.
- Classify a function as even, odd or neither, and state the domain and range of a transformed function.
- Evaluate a piecewise or step function and match a description to its rule.
- Find the average rate of change of a function over an interval from a rule, a table or a graph.
- Build a polynomial or rational model from a described situation and use it to answer a question.
- Choose the function family a situation calls for, interpret its parameters, and solve direct, inverse and joint variation problems.