Graph of a function; sketching with technology practice questions.

Let f(x) = x² − 3x. Determine which of the following points lies on the graph of y = f(x).

• (2, 2)
• (2, −2)
• (−1, −4)
• (3, 9)

Answer: (2, −2)

A point (a, b) lies on the graph exactly when f(a) = b. Here f(2) = 4 − 6 = −2, so (2, −2) is on the graph.

The point (a, b) lies on the graph of y = f(x). State which condition must be satisfied.

• f(a) = b
• f(b) = a
• a = b
• f(a) = f(b)

Answer: f(a) = b

The graph of y = f(x) consists of all points (x, f(x)), so (a, b) is on the graph precisely when f(a) = b. Swapping input and output is a classic error.

Determine which of the following graphs never goes below the x-axis.

• y = x³
• y = x − 2
• y = x²
• y = x² − 1

Answer: y = x²

x² ≥ 0 for all real x, so the graph of y = x² touches the x-axis at the origin but never goes below it. Subtracting 1 shifts the graph down, so y = x² − 1 does dip below.

The depth of water at a harbour rises and falls with the tide, repeating the same pattern about every 12 hours. Suggest which type of function is most appropriate for sketching depth against time.

• Linear
• Quadratic
• Exponential
• Sinusoidal

Answer: Sinusoidal

A repeating rise-and-fall pattern is periodic, so a sinusoidal (sine or cosine) model matches the context; the other models are not periodic.

Let f(x) = x², g(x) = 2x and h(x) = f(x) + g(x). Calculate h(3).

• 9
• 15
• 6
• 54

Answer: 15

h(3) = f(3) + g(3) = 9 + 6 = 15. Using only f(3) = 9 or only g(3) = 6, or multiplying instead of adding, gives the distractors.

A ball is thrown vertically upwards and falls back to the ground. Suggest the shape of the graph of its height against time.

• A straight line with negative gradient
• A parabola opening downwards
• A parabola opening upwards
• An increasing exponential curve

Answer: A parabola opening downwards

Height increases, reaches a maximum, then decreases symmetrically under constant gravity — a downward-opening parabola.