Foundations: Advanced Math

Advanced Math is worth the same thirteen to fifteen questions as Algebra, and its name scares students more than its content should. "Advanced" on the SAT means nonlinear: quadratics, exponentials, polynomials, radicals and rational expressions. Every one of those rests on a short list of mechanical skills that this lesson teaches: exponent rules, function notation, factoring, and reading a parabola.

If Algebra is about lines, Advanced Math is about curves, and the first thing to learn is that a curve's equation and its graph say the same things in two languages. A quadratic's factors are its x-intercepts; its constant term is its y-intercept; the sign of its leading coefficient is whether it opens up or down. An exponential's base is whether it grows or decays and its coefficient is where it starts. Translating between the equation and the picture is most of the domain.

The Medium lesson adds the quadratic formula, vertex form, exponential models and rational expressions; the Advanced lesson adds parameters, sums and products of roots, and equivalent-expression questions with unknown constants. Both assume you can do everything here quickly.

Exponents are repeated multiplication, and the rules follow from that. Multiplying powers with the same base adds the exponents: x^a times x^b = x^(a+b). Dividing subtracts them: x^a / x^b = x^(a-b). Raising a power to a power multiplies them: (x^a)^b = x^(ab). A power of a product distributes to each factor: (xy)^a = x^a y^a. Any nonzero base to the zero power is 1, and a negative exponent means a reciprocal: x^(-a) = 1/x^a. The rules never combine different bases, so 2³ times 3² is just 8 times 9 = 72. Radicals are exponents in another notation: √x is x^(1/2), and the Medium lesson develops rational exponents from there.

A function is a rule that assigns exactly one output to each input. The notation f(x) names the rule f and the input x; f(x) = 2x² - 3x + 1 says the output is found by squaring the input, doubling, subtracting three times the input and adding one. Evaluating means substituting: f(2) = 2(4) - 6 + 1 = 3, and f(-1) = 2(1) + 3 + 1 = 6, where the parentheses around -1 are what keep the signs right. A table of values or a graph defines a function just as well as a formula; on a graph, f(3) is the y-coordinate of the point whose x-coordinate is 3, and solving f(x) = 0 means finding where the graph crosses the x-axis.

Factoring rewrites a sum as a product, and it is the main tool for solving nonlinear equations. First factor out the greatest common factor of every term: 6x² + 9x = 3x(2x + 3). Second, look for a difference of squares, which factors as a² - b² = (a - b)(a + b); so x² - 49 = (x - 7)(x + 7) and 4x² - 25 = (2x - 5)(2x + 5). Third, factor a trinomial of the form x² + bx + c by finding two numbers whose product is c and whose sum is b: for x² + 5x + 6 the numbers are 2 and 3, so it factors as (x + 2)(x + 3); for x² - 5x + 6 they are -2 and -3. The special cases x² + 2ab x + b² = (x + b)² and x² - 2bx + b² = (x - b)² are perfect square trinomials. Check any factoring by multiplying back out with FOIL: first, outer, inner, last.

A quadratic equation, ax² + bx + c = 0, has up to two real solutions. When the quadratic factors, the zero product property finishes it: a product is zero only when a factor is zero, so set each factor equal to zero and solve. For x² - 5x + 6 = 0, factoring gives (x - 2)(x - 3) = 0 and the solutions are x = 2 and x = 3. If the equation is not already equal to zero, move everything to one side first. Quadratics that do not factor nicely are solved with the quadratic formula, in the Medium lesson; at the Foundations level every quadratic factors over the integers.

The graph of y = ax² + bx + c is a parabola, a symmetric U-shaped curve. If a is positive the parabola opens upward and its vertex is its lowest point; if a is negative it opens downward and the vertex is the highest point. The y-intercept is (0, c), since substituting x = 0 leaves only c. The x-intercepts, also called zeros or roots, are the real solutions of ax² + bx + c = 0, so the factored form reveals them at once: y = (x + 1)(x - 3) crosses the x-axis at -1 and 3. The axis of symmetry is the vertical line halfway between the x-intercepts, at x = -b/(2a), and the vertex sits on it; its y-coordinate is found by substituting that x back in. For y = x² - 2x - 3 = (x + 1)(x - 3), the axis is x = 1 and the vertex is (1, -4).

An exponential function has the form f(x) = a times b^x with a nonzero and b positive and not 1. The coefficient a is the starting value, f(0), because b⁰ = 1. The base b is the growth factor: if b is greater than 1 the function grows, multiplying by b for each unit increase in x; if b is between 0 and 1 it decays. A quantity that increases 20 percent per year has b = 1.20; one that loses 20 percent per year has b = 0.80. The difference between linear and exponential is the difference between adding the same amount each step and multiplying by the same factor each step; a table whose outputs have a constant ratio, not a constant difference, is exponential. Exponential graphs never cross the x-axis when a is positive and they pass through (0, a).

Combining different bases: 2³ times 3² is 72, not 6⁵.

Forgetting to raise the coefficient to the power: (3x²)³ is 27x⁶, not 3x⁶.

Reading f(4) as f times 4, or squaring a negative input without parentheses so that (-3)² becomes -9.

Factoring a trinomial without pulling out the greatest common factor first, leaving a factor trapped inside.

Setting factors equal to a nonzero number: (x - 2)(x + 5) = 8 does not mean x - 2 = 8. Get zero on one side first.

Mixing up the sign of the x-intercepts: (x + 5) is zero at x = -5.

Writing a 4 percent growth base as 0.04 or 4 instead of 1.04, and a 15 percent decay base as 0.15 instead of 0.85.