549 exam-style questions on Functions with answers and explanations. Six real samples below — the full set is free with an account.
What is the equation of the vertical asymptote of f(x) = (2x + 1)/(x − 3)?
Answer: x = 3
Vertical asymptotes of rational functions occur where the denominator is zero (and the numerator is not): x − 3 = 0 gives x = 3. The horizontal asymptote here would be y = 2.
The graph of y = f(2x) is obtained from y = f(x) by which single transformation?
Answer: horizontal stretch, scale factor 1/2
Replacing x by 2x compresses the graph horizontally: every point (a, b) maps to (a/2, b). Transformations inside the bracket act horizontally and 'in reverse'.
The graph of y = f(x − 3) is obtained from the graph of y = f(x) by a translation of
Answer: 3 units right
Changes inside the bracket act horizontally and in the opposite direction to the sign: x − 3 shifts the graph 3 units right.
The vertex of the parabola y = x² − 6x + 11 is at
Answer: (3, 2)
Complete the square: y = (x − 3)² + 2, so the vertex is (3, 2). Check: x = −b/(2a) = 3 and y(3) = 9 − 18 + 11 = 2.
The horizontal asymptote of the graph of y = 5e⁻ˣ + 2 is
Answer: y = 2
As x → ∞, e⁻ˣ → 0, so y → 2. The “+2” shifts the natural asymptote y = 0 up to y = 2.
The gradient of the straight line through the points (1, 2) and (3, 10) is
Answer: 4
m = (10 − 2) / (3 − 1) = 8 / 2 = 4.
That was 6 of 549 Functions questions.
Practise all 549 freeA student writes the equation of the line through (−2, 5) with gradient 4 as y + 5 = 4(x − 2).
See the correction →A student writes x² + 6x + 1 in the form (x + 3)² + 1.
See the correction →A student solves (x + 2)/x ≥ 1 by multiplying both sides by x, obtaining x + 2 ≥ x, and concludes that every real x ≠ 0 is a solution.
See the correction →549 Functions questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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