Advanced Math

~35% of the section. Quadratics, exponentials, polynomials, and function notation.

2 min read

The other 35%, and where most of the score ceiling lives. If you want a 700+, this is the folder to over-invest in.

#Quadratics — three forms, three uses

Form	Reads off
Standard `ax² + bx + c`	y-intercept is c
Factored `a(x − p)(x − q)`	roots are p and q
Vertex `a(x − h)² + k`	vertex is (h, k)

The test asks "which form best shows the ___" constantly. The answer is always the form that has the thing sitting in it as a number.

The quadratic formula is not on the reference sheet: x = (−b ± √(b² − 4ac)) / 2a. Memorise it.

The discriminant b² − 4ac gives the number of real solutions without solving:

> 0 → two real solutions
= 0 → exactly one
< 0 → none

Questions asking "for what value of k does this have exactly one solution" are discriminant questions. Set it to zero.

Vertex x-coordinate = −b/2a. That's the axis of symmetry, the maximum or minimum, and the answer to most "what value of x maximises..." questions.

#Exponents and radicals

xᵃ · xᵇ = xᵃ⁺ᵇ
xᵃ / xᵇ = xᵃ⁻ᵇ
(xᵃ)ᵇ = xᵃᵇ
x⁻ᵃ = 1/xᵃ
x^(a/b) = ᵇ√(xᵃ)
x⁰ = 1

Exponential growth and decay: y = a(1 ± r)ᵗ — a is the initial amount, r the rate per period, t the number of periods. If the period isn't 1 (compounded monthly over t years), it's a(1 + r/n)^(nt).

Linear vs. exponential is a recurring question: linear changes by a constant amount per step; exponential changes by a constant factor. Check a table by subtracting consecutive values (constant → linear) and then dividing them (constant → exponential).

#Polynomials

Factor theorem: if p(a) = 0 then (x − a) is a factor, and a is an x-intercept. These three statements are the same statement, and the test rotates between them.
Remainder theorem: dividing p(x) by (x − a) leaves remainder p(a).
End behaviour is set by the leading term's degree and sign.

#Function notation

f(3) means substitute 3 for x. f(g(x)) means evaluate g first, then feed the result into f. Work inside out, and write the intermediate value down — composed-function errors are almost all bookkeeping.

Transformations of f(x):

f(x) + k → up k
f(x + k) → left k (not right — this is the one everyone reverses)
−f(x) → flip over the x-axis
f(−x) → flip over the y-axis

#Rational equations

Multiply through by the denominator, solve, then check for extraneous solutions — any value that makes an original denominator zero is not a solution, however cleanly it fell out of the algebra. The test includes it as a wrong answer every time.