Functions practice questions.

What is the equation of the vertical asymptote of f(x) = (2x + 1)/(x − 3)?

• x = 3
• x = −3
• y = 2
• x = −1/2

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?

• horizontal stretch, scale factor 2
• horizontal stretch, scale factor 1/2
• vertical stretch, scale factor 2
• vertical stretch, scale factor 1/2

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

• 3 units left
• 3 units right
• 3 units up
• 3 units down

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

• (3, 2)
• (−3, 2)
• (3, −2)
• (6, 11)

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

• y = 0
• y = 2
• y = 5
• y = 7

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

• 2
• 4
• 6
• 8

Answer: 4

m = (10 − 2) / (3 − 1) = 8 / 2 = 4.