SAT (Scholastic Assessment Test) Math: Advanced Math Syllabus

What are the main methods for solving a quadratic equation?

Factoring, using the quadratic formula, completing the square, taking square roots, and graphing to find x-intercepts.

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

If a·b = 0, then a = 0 or b = 0. After factoring a quadratic to 0, set each factor equal to 0 and solve.

How do you solve x² = 49 by taking square roots?

x = ±√49 = ±7. Remember both the positive and negative roots.

What is the process of completing the square for x² + bx?

Add (b/2)²: x² + bx + (b/2)² = (x + b/2)². You add and subtract (b/2)² to keep the equation balanced.

What is the discriminant, and what is its formula?

The discriminant is b² − 4ac, the expression under the radical in the quadratic formula. It determines the number and type of solutions.

If the discriminant b² − 4ac > 0, how many real solutions does the quadratic have?

Two distinct real solutions (the parabola crosses the x-axis at two points).

If the discriminant b² − 4ac = 0, how many real solutions are there?

Exactly one real solution (a repeated/double root); the parabola is tangent to the x-axis.

If the discriminant b² − 4ac < 0, how many real solutions are there?

No real solutions (two complex solutions); the parabola does not cross the x-axis.

What is the standard form of a parabola (quadratic function)?

y = ax² + bx + c, where c is the y-intercept and a controls direction/width.

What is the vertex form of a parabola, and what does it reveal?

y = a(x − h)² + k, where (h, k) is the vertex. It directly shows the maximum or minimum point.

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

y = a(x − p)(x − q), where p and q are the x-intercepts (roots/zeros).

How do you find the x-coordinate of a parabola's vertex from standard form?

x = −b/(2a). This is also the axis of symmetry.

What does the sign of 'a' tell you about a parabola?

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

What is the axis of symmetry of a parabola?

The vertical line x = −b/(2a) (or x = h in vertex form) that passes through the vertex and mirrors the two sides.

In a quadratic model like h = −16t² + v₀t + h₀ for projectile height, what does the vertex represent?

The vertex gives the maximum height reached and the time at which it occurs.

In a quadratic revenue/profit model, what do the x-intercepts typically represent?

The zeros/break-even points where the quantity (revenue or profit) equals zero.

What is the general form of an exponential function?

y = a·b^x, where a is the initial value (y-intercept) and b is the constant growth/decay factor (base).

In y = a·b^x, how do you tell growth from decay?

If b > 1 it's exponential growth; if 0 < b < 1 it's exponential decay.

How do you write the growth factor for an exponential increasing by r percent each period?

b = 1 + r (as a decimal), so y = a(1 + r)^x. For 5% growth, b = 1.05.

How do you write the decay factor for an exponential decreasing by r percent each period?

b = 1 − r (as a decimal), so y = a(1 − r)^x. For 5% decay, b = 0.95.

What is the key difference between linear and exponential change?

Linear changes by a constant amount (addition) each step; exponential changes by a constant factor (multiplication/percent) each step.

What does function notation f(x) mean, and how do you evaluate f(3)?

f(x) is the output of function f for input x. To find f(3), substitute 3 for every x in the function and simplify.

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

It shifts the graph vertically: up by k if k > 0, down by k if k < 0.

How does f(x + h) transform the graph of f(x)?

It shifts the graph horizontally: left by h if h > 0, right by h if h < 0 (opposite of the sign inside).

How does −f(x) versus f(−x) transform a graph?

−f(x) reflects over the x-axis (flips vertically); f(−x) reflects over the y-axis (flips horizontally).

How does a·f(x) with a > 1 versus 0 < a < 1 affect a graph?

a > 1 stretches the graph vertically (taller/narrower); 0 < a < 1 compresses it vertically (shorter/wider).

What does the end behavior of a polynomial depend on?

Its leading term: the degree (even vs odd) and the sign of the leading coefficient determine the direction of both ends.

What is a vertical asymptote of a rational function and how do you find it?

A vertical line the graph approaches but never crosses; it occurs where the denominator equals zero (after simplifying) but the numerator does not.

How do you typically solve a nonlinear system like y = x² and y = x + 2?

Use substitution: set the expressions equal (x² = x + 2), rearrange to x² − x − 2 = 0, solve for x, then find y. Solutions are the intersection points of the graphs.