Free HSC Mathematics Extension 1 Flashcard Review
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Solving rational inequalities
Move to one side, common denominator, factor, sign table.
Solving |f(x)| > a
f(x) > a OR f(x) < -a when a >= 0.
Solving |f(x)| < a
-a < f(x) < a when a > 0.
Solving |f(x)| = |g(x)|
f(x) = g(x) OR f(x) = -g(x).
Sketching y = 1 / f(x)
Vertical asymptotes where f(x) = 0.
Sketching y = |f(x)|
Reflect negatives about x-axis. Always nonnegative.
Sketching y = f(|x|)
Even function. Symmetric about y-axis.
Sketching y = √(f(x))
Domain: f(x) ≥ 0. Graph above x-axis.
Parametric curves
x = x(t), y = y(t), t in [a, b].
Sketching parametric curves
Table of values, plot points, show direction.
Parametric gradient
dy/dx = (dy/dt) / (dx/dt).
Polynomial division
Divide leading terms, multiply, subtract, repeat.
Remainder Theorem
Remainder when P(x) divided by (x - a) is P(a).
Factor Theorem
(x - a) is factor of P(x) iff P(a) = 0.
Sum and product roots quadratic
ax2 + bx + c = 0: sum = -b/a, product = c/a.
Sum and product roots cubic
ax3 + bx2 + cx + d = 0: sum = -b/a, product = -d/a.
Domain and range sin-1 x
Domain [-1, 1], range [-pi/2, pi/2].
Domain and range cos-1 x
Domain [-1, 1], range [0, pi].
Domain and range tan-1 x
Domain R, range (-pi/2, pi/2).
sin(A + B) formula
sin A cos B + cos A sin B.