Free HSC Year 11 Mathematics Advanced Flashcard Review
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What does a^(m/n) mean?
(ⁿ√a)ᵐ, the nth root of a raised to the power m
Simplify a⁻ⁿ
1/aⁿ
How do you rationalise 1/(a + √b)?
Multiply top and bottom by the conjugate a − √b
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a
Meaning of Δ < 0, Δ = 0, Δ > 0
No real roots; one repeated root; two distinct real roots
Vertical line test
A graph is a function if every vertical line meets it at most once
Natural domain of √(x − 4)
x ≥ 4
Point–gradient formula
y − y₁ = m(x − x₁)
Condition for perpendicular lines
m₁m₂ = −1
Axis of symmetry of y = ax² + bx + c
x = −b/2a
Asymptotes of y = k/x
x = 0 and y = 0
Inverse variation model
y = k/x, found by substituting one data pair to get k
What is the equation of a circle centre O, radius r, and where does it come from?
x² + y² = r², from Pythagoras' theorem applied to the right triangle with legs |x|, |y| and hypotenuse r.
Radius of x² + y² = 45?
r = √45 = 3√5.
Domain and range of y = √(r² − x²)?
Domain [−r, r]; range [0, r] (upper semicircle).
Which semicircles are functions?
Only the upper y = √(r² − x²) and lower y = −√(r² − x²) semicircles.
Write 'x is greater than −1 and at most 6' in interval notation.
(−1, 6]
Algebraic test for an even function?
f(−x) = f(x) for all x in the domain; graph symmetric about the y-axis.
Algebraic test for an odd function?
f(−x) = −f(x) for all x in the domain; point symmetry about the origin.
If f(x) = x + 1 and g(x) = 3x, what is f(g(x))?
f(3x) = 3x + 1.