HSC Year 11 Mathematics Extension 1 practice exams with worked answers

Extension 1 asks you to build new graphs out of a graph you already know. The first and most important transformation is the reciprocal: given y = f(x), sketch y = 1/f(x). Rather than plotting points, you read features straight off the original curve.

• Zeros become vertical asymptotes. Wherever f(x) = 0, the reciprocal is undefined and its magnitude grows without bound, so draw a dashed vertical line there.
• Large values become small values. If f(x) → ±∞, then 1/f(x) → 0. In particular, if f(x) grows without bound as x → ±∞, the reciprocal has the horizontal asymptote y = 0. If f(x) → k (k ≠ 0), then 1/f(x) → 1/k.
• Sign is preserved. 1/f(x) is positive exactly where f(x) is positive, so the reciprocal stays on the same side of the x-axis.
• The values 1 and −1 are fixed. Where f(x) = ±1, the two graphs meet. These are useful anchor points.
• Turning points stay at the same x-value. A local maximum of f with a non-zero value becomes a local minimum of 1/f, and vice versa, because a larger positive number has a smaller reciprocal.
• Increasing becomes decreasing on any interval where f does not change sign.

Reciprocal trigonometric functions. The three reciprocal ratios are cosec x = 1/sin x, sec x = 1/cos x and cot x = 1/tan x = cos x / sin x. Apply the rules above to the familiar sine, cosine and tangent graphs.

• y = cosec x has vertical asymptotes where sin x = 0, that is x = nπ (or 180°n). It touches y = 1 at x = π/2 + 2nπ and y = −1 at x = 3π/2 + 2nπ. Range: y ≤ −1 or y ≥ 1. Period 2π.
• y = sec x has asymptotes where cos x = 0, that is x = π/2 + nπ. It has local minima of 1 at x = 2nπ and local maxima of −1 at x = π + 2nπ. Range: y ≤ −1 or y ≥ 1. Period 2π. It is an even function, like cosine.
• y = cot x has asymptotes at x = nπ, zeros at x = π/2 + nπ, is decreasing on each branch and has period π. Its range is all real numbers.

When working in degrees the same shapes apply, with asymptotes at multiples of 90° or 180°. Always label asymptotes with dashed lines and mark the points where the reciprocal meets the original curve at y = ±1.