PSAT/NMSQT (Preliminary SAT/National Merit Scholarship Qualifying Test) Math: Advanced Math Syllabus

How do you rationalize a denominator containing a single square root, like 1/√a?

Multiply numerator and denominator by √a to get √a / a.

State the quadratic formula for ax² + bx + c = 0.

x = (−b ± √(b² − 4ac)) / (2a).

What is the discriminant, and what does its sign tell you?

The discriminant is b² − 4ac. Positive → two real solutions; zero → one real (repeated) solution; negative → no real solutions (two complex).

What does the Zero Product Property say, and how is it used to solve quadratics?

If AB = 0 then A = 0 or B = 0. After factoring a quadratic to 0, set each factor equal to zero and solve.

Describe how to solve a quadratic by completing the square.

Move the constant aside, make the leading coefficient 1, add (b/2)² to both sides to form a perfect square, then take the square root of both sides and solve.

How do you solve a radical equation such as √(x) = c?

Isolate the radical, square both sides to eliminate it, solve the resulting equation, then check for extraneous solutions.

Why must you check solutions when solving radical equations?

Squaring both sides can introduce extraneous solutions that satisfy the squared equation but not the original.

How do you solve a rational equation?

Multiply both sides by the least common denominator to clear fractions, solve the resulting equation, and reject any solution that makes a denominator zero.

What are extraneous solutions in a rational equation?

Values obtained algebraically that make an original denominator zero (undefined), so they must be discarded.

How do you solve an exponential equation when both sides can be written with the same base?

Rewrite both sides with a common base, set the exponents equal, and solve. E.g., 2^x = 8 → 2^x = 2³ → x = 3.

How do you solve an exponential equation using logarithms, like 3^x = 20?

Take the log of both sides: x = log(20)/log(3), since log(3^x) = x·log(3).

What is the standard form of a quadratic function?

f(x) = ax² + bx + c, where a ≠ 0.

What is the vertex form of a quadratic function and what does it reveal?

f(x) = a(x − h)² + k, where (h, k) is the vertex of the parabola.

What is the formula for the x-coordinate of a parabola's vertex (axis of symmetry) from standard form?

x = −b / (2a).

What does the sign of 'a' tell you about a parabola y = ax² + bx + c?

If a > 0 the parabola opens upward (vertex is a minimum); if a < 0 it opens downward (vertex is a maximum).

What is the factored (intercept) form of a quadratic and what does it reveal?

f(x) = a(x − p)(x − q); p and q are the x-intercepts (roots/zeros) of the parabola.

How can you find the y-intercept of a quadratic function f(x) = ax² + bx + c?

Evaluate f(0), which equals c; the y-intercept is (0, c).

What is the general form of an exponential function?

f(x) = a·bˣ, where a is the initial value (y-intercept) and b > 0, b ≠ 1, is the base (growth/decay factor).

How do you tell exponential growth from exponential decay in f(x) = a·bˣ?

With a > 0: if b > 1 it is growth; if 0 < b < 1 it is decay.

In f(x) = a·bˣ, what does the value 'a' represent?

The initial value, i.e., the y-intercept (the value of f when x = 0).

What is the horizontal asymptote of the basic exponential function f(x) = a·bˣ?

The x-axis, y = 0.

How do you find the percent rate of change from an exponential function f(x) = a·bˣ?

The rate is (b − 1)·100%. Growth: b = 1 + r; decay: b = 1 − r, where r is the decimal rate.

What does function notation f(x) mean?

f(x) is the output (value) of the function f when the input is x; it is read 'f of x.'

How do you evaluate a composite function f(g(x))?

First compute g(x), then substitute that result into f as its input.

How does the graph of f(x) + k compare to f(x)?

It is a vertical shift: up by k if k > 0, down by |k| if k < 0.

How does the graph of f(x − h) compare to f(x)?

It is a horizontal shift: right by h if h > 0, left by |h| if h < 0.

What transformation does −f(x) versus f(−x) produce?

−f(x) reflects the graph across the x-axis; f(−x) reflects it across the y-axis.

What does the factor 'a' do in a·f(x) when |a| > 1 versus 0 < |a| < 1?

|a| > 1 stretches the graph vertically; 0 < |a| < 1 compresses it vertically (and a < 0 also reflects it across the x-axis).

How do you solve a system with one linear and one nonlinear equation algebraically?

Use substitution: solve the linear equation for one variable, substitute into the nonlinear equation, solve, then back-substitute to find the other variable.

How many solutions can a system of a line and a parabola have, and what do they represent graphically?

Zero, one, or two solutions, corresponding to the points where the line and parabola intersect (no intersection, tangent, or two crossings).