Free HSC Mathematics Advanced Flashcard Review
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Define a function transformation: vertical translation by k units
If y = f(x), then y = f(x) + k shifts the graph vertically by k units (up if k > 0, down if k < 0).
Define a function transformation: horizontal translation by h units
If y = f(x), then y = f(x − h) shifts the graph horizontally by h units (right if h > 0, left if h < 0).
Define a function transformation: vertical dilation by factor a
If y = f(x), then y = a·f(x) stretches the graph vertically by factor |a| (or compresses if |a| < 1). Reflection across x-axis if a < 0.
Define a function transformation: horizontal dilation by factor 1/b
If y = f(x), then y = f(bx) compresses the graph horizontally by factor |1/b| (or stretches if |b| < 1). Reflection across y-axis if b < 0.
State the key features of the reciprocal function y = 1/x
Vertical asymptote: x = 0; horizontal asymptote: y = 0; domain: x ≠ 0; range: y ≠ 0; hyperbola in quadrants I and III; odd function.
State the key features of y = 1/(x−a) + b
Vertical asymptote: x = a; horizontal asymptote: y = b; domain: x ≠ a; range: y ≠ b; centre of hyperbola at (a, b).
Define the absolute value function and state its graph properties
y = |f(x)| reflects all negative values of f(x) to positive. Graph is always y ≥ 0; corners occur where f(x) = 0. V-shaped or piecewise linear appearance.
State the domain and range of y = √(x)
Domain: x ≥ 0; range: y ≥ 0. Square root function; always increasing; concave down.
Define a combined function (composite function) f∘g
f∘g(x) = f(g(x)); the output of g becomes the input of f. Domain of f∘g: {x in domain of g : g(x) in domain of f}.
State the period of y = sin(kx) or y = cos(kx)
Period = 2π/|k|. For y = sin(kx) or y = cos(kx), the function completes one full cycle in interval 2π/|k|.
State the amplitude of y = a·sin(x) or y = a·cos(x)
Amplitude = |a|. The graph oscillates between −|a| and |a|. Amplitude is always non-negative.
State the phase shift of y = sin(x − c) or y = cos(x − c)
Phase shift = c units to the right (if c > 0) or |c| units to the left (if c < 0). General form: y = a·sin(k(x − c)) + d has phase shift c/k.
State the general form of a sinusoidal function
y = a·sin(k(x − c)) + d or y = a·cos(k(x − c)) + d, where |a| = amplitude, period = 2π/k, phase shift = c, vertical shift = d.
List the exact values: sin(0°), sin(30°), sin(45°), sin(60°), sin(90°)
sin(0°) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1.
List the exact values: cos(0°), cos(30°), cos(45°), cos(60°), cos(90°)
cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0.
List the exact values: tan(0°), tan(30°), tan(45°), tan(60°)
tan(0°) = 0, tan(30°) = 1/√3 = √3/3, tan(45°) = 1, tan(60°) = √3.
State the ASTC rule for trigonometric signs in quadrants
Quadrant I (0° to 90°): All positive. Quadrant II (90° to 180°): Sin positive. Quadrant III (180° to 270°): Tan positive. Quadrant IV (270° to 360°): Cos positive.
State the Pythagorean identity
sin2(θ) + cos2(θ) = 1 for all θ. Derived from x2 + y2 = r2 on the unit circle.
State the angle addition formula: sin(A + B)
sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
State the angle addition formula: cos(A + B)
cos(A + B) = cos(A)cos(B) − sin(A)sin(B).