Advanced Math

Absolute value can model distance on a number line. This viewpoint makes solution counts and tolerance statements easier to interpret.

An absolute value equation with a positive right side usually splits into two linear equations. A zero right side gives one solution, and a negative right side gives no solution.

Absolute value inequalities describe distances. Less-than inequalities produce inside intervals; greater-than inequalities produce outside intervals.

Equivalent exponential expressions use exponent rules, such as adding exponents when multiplying powers with the same base and multiplying exponents when raising a power to a power. Rewriting expressions with a common base is often the key step.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, and simplifying rational expressions while respecting restrictions on denominators.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Parameter questions for equivalent expressions often ask for missing coefficients or constants after expanding, factoring, completing the square, or simplifying rational expressions.

Equivalent-expression questions often require factoring before canceling, applying exponent rules, or simplifying radicals. Restrictions such as $x\ne4$ matter because canceling a factor does not make the original expression defined at the canceled value.

Polynomial expressions can often be simplified or rewritten by distributing, combining like terms, factoring out common factors, and recognizing special products such as differences of squares.

Quadratic expressions can be written in standard form, factored form, or vertex form. Moving between these forms reveals different information, such as zeros, coefficients, or the vertex.

Radical and rational expression questions require precise use of factoring, cancellation, exponent rules, and domain restrictions. Equivalent expressions must match the original expression on its allowed domain.

Function notation asks for outputs from specified inputs or for inputs that produce specified outputs. Transformations outside $f$ change outputs; transformations inside $f$ change inputs.

Exponential equations on the SAT often become linear equations after rewriting both sides with the same base. When bases match, the exponents can be set equal.

Nonlinear equations in one variable include quadratic, radical, rational, absolute value, exponential, and polynomial equations. Solving often requires factoring, taking square roots, isolating a radical, using exponent rules, or checking for restrictions and extraneous solutions.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms.

Parameter questions for nonlinear equations often use roots, discriminants, squared forms, radical equations, or factored forms.

Polynomial equations are often solved by moving all terms to one side, factoring, and applying the zero product property. Some SAT-style items ask for a particular solution, a nonzero solution, or the number of real solutions shown by factored form.

Quadratic equations may be solved by factoring, taking square roots, or using the discriminant to reason about the number of real solutions. SAT-style items often ask for one solution, a parameter, or the number of real solutions rather than both roots.

Radical and rational equations are common sources of errors because restrictions and extraneous solutions must be checked. Squaring and clearing denominators can introduce invalid candidates.

Radical and rational equations require attention to restrictions. Rational equations can have excluded denominator values, and radical equations can produce extraneous solutions after squaring.

Exponential functions such as $f(x)=a(b)^x$ are interpreted using the initial value $a$ and the growth or decay factor $b$. Values greater than 1 indicate growth, while values between 0 and 1 indicate decay.

Nonlinear function questions may involve transformed functions, vertices, zeros, symmetry, evaluating composite-looking expressions, or comparing function features from different forms.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Parameter questions for nonlinear functions may use vertex form, domain restrictions, exponential factors, or input-output conditions.

Function questions involving radicals and rational expressions often ask for a value, an excluded input, or an endpoint of the domain. The key restrictions are that square-root radicands must be nonnegative and denominators cannot equal zero.

Polynomial function questions often use factored form to identify zeros and standard form to identify intercepts or evaluate function values. Parameter questions may ask for a coefficient or constant from a known zero or value.

Quadratic functions can be interpreted through vertex, standard, and factored forms. Vertex form gives the vertex and maximum or minimum value, while factored form gives the zeros and the axis of symmetry.

Radical and rational functions often ask for domain restrictions, excluded inputs, and endpoint values. Square-root radicands must be nonnegative, and denominators cannot equal zero.

Radical and rational equations often produce candidates that must be checked. Squaring can introduce extraneous solutions, and denominators cannot be zero.

Systems involving a line and a parabola can have zero, one, or two real solutions. Setting the equations equal usually creates a quadratic equation, and the discriminant determines the number of intersections.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Parameter questions for nonlinear systems often use a line-parabola system, root relationships, tangency, or identities involving sums and products.

A line-parabola system can be solved by setting the two expressions for $y$ equal. The resulting quadratic tells how many intersection points exist.