SAT Advanced Math practice questions
Quadratics, exponentials, polynomials and nonlinear systems: the questions that separate strong Math scores from great ones.
35 free questions, each with a full worked solution and an explanation of what each wrong answer means.
Practise this domain adaptively →Easy (6)
- Evaluating linear functions
If f(x) = 3x − 7, what is the value of f(5)? - Difference of squares
What are the solutions to x² − 9 = 0? - Factoring quadratic expressions
Which of the following expressions is equivalent to x² + 7x + 12 ? - Evaluating a polynomial function at a negative input
The function g is defined by g(x) = x³ + 2x. What is the value of g(−2) ? - Rational exponents and radical form
For x > 0, which of the following is equivalent to ∛(x⁵) ? - Zeros of a quadratic given in factored form
The function h is defined by h(x) = (x − 6)(x + 2). What is one value of x for which h(x) = 0 ?
Easy–Medium (7)
- Function notation
If f(x) = 2x² − 3, what is f(4)? - Exponent rules
Which expression is equivalent to (2x³)(3x⁴)? - Difference of squares
What are the solutions to x² − 16 = 0? - Expanding a squared binomial
Which expression is equivalent to (x + 5)²? - Simplifying rational expressions
For x ≠ 0, which expression is equivalent to (x² + 3x) / x? - Solving a quadratic equation by taking square roots
What are the solutions to the equation 2(x − 3)² − 32 = 0 ? - Using a known zero of a quadratic to find an unknown coefficient
The function h is defined by h(x) = x² + kx + 9, where k is a constant. If h(3) = 0, what is the value of k ?
Medium (7)
- Quadratic equations
What are the solutions to x² − 5x + 6 = 0? - Vertex form of a parabola
The graph of f(x) = (x − 3)² + 4 is a parabola. What are the coordinates of its vertex? - Factoring polynomials
If x² − 2x − 8 = (x + 2)(x − a), what is the value of a? - Radical equations
If √(x + 5) = 4, what is the value of x? - Composition of functions
If f(x) = x + 3 and g(x) = 2x, what is f(g(4))? - Solving quadratics by factoring
What are the solutions to x² + 6x + 8 = 0? - Systems with a nonlinear equation
The system y = x² and y = x + 6 has two solutions. What is the sum of the two x-values of those solutions?
Hard (8)
- Exponential functions
A population of bacteria doubles every 3 hours. If it starts at 500, which function models the population P after t hour… - Discriminant and real solutions
How many real solutions does x² + 4x + 5 = 0 have? - Exponential decay
A car worth $20,000 loses 15% of its value each year. What is its value after 2 years? - Exponential growth modeling
An investment of $1,000 grows at 6% per year, compounded annually. Which expression gives its value after t years? - Composition of functions
If f(x) = x² − 1 and g(x) = x + 2, what is g(f(3))? - Vertex and axis of symmetry
The parabola y = x² − 8x + 12 crosses the x-axis at two points. What is the x-coordinate of the vertex? - Radical equations and extraneous solutions
What is the solution set of the equation x − √(x + 5) = 1 ? - Solving exponential equations by rewriting with a common base
If (1/8)^x · 4^(2x + 1) = 16^(x − 3), what is the value of x ?
Very hard (7)
- Converting to vertex form
The function f(x) = x² + 6x + 5 is rewritten in the form f(x) = (x + a)² + b. What is the value of b? - Solving rational equations
What is the solution to the equation 6/(x − 2) = 3? - Zeros of a polynomial
The polynomial p(x) = x³ − 4x has how many distinct real zeros? - Polynomial identities: factor theorem and remainder theorem with unknown coefficients
The polynomial p is defined by p(x) = x³ + ax² + bx − 12, where a and b are constants. If x − 2 is a factor of p(x), and… - Rewriting exponential functions to reveal a growth rate over a different time interval
The function f, defined by f(x) = 3000(1.331)^(x/12), gives the value, in dollars, of an investment x months after it wa… - Tangency condition for a line and a parabola (discriminant with a parameter)
In the xy-plane, the parabola y = x² − 4x + 7 and the line y = mx − 2, where m is a constant, intersect at exactly one p… - Determining a quadratic from its zeros and extreme value, then evaluating it
The function f is defined by f(x) = ax² + bx + c, where a, b, and c are constants. If f(2) = f(8) = 0 and the minimum va…
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