Graph of a function; sketching; graphing with technology practice questions.

A quadratic graph opens downwards and has its maximum at (3, 8). State the equation of its axis of symmetry.

• x = 3
• y = 8
• x = 8
• y = 3

Answer: x = 3

The axis of symmetry of a parabola is the vertical line through its vertex, so x = 3. A common slip is confusing it with the maximum value y = 8.

Using technology, the graphs of y = x³ − 2x and y = x intersect. Determine the number of points of intersection.

• 3
• 1
• 2
• 0

Answer: 3

Solve x³ − 2x = x → x³ − 3x = 0 → x(x² − 3) = 0, giving x = 0, √3, −√3: three intersection points.

The graph of an exponential model f(x) = 5(2ˣ) has a horizontal asymptote. State its equation.

• y = 0
• y = 5
• y = 2
• x = 0

Answer: y = 0

As x → −∞, 2ˣ → 0 so f(x) → 0; the horizontal asymptote is y = 0. The factor 5 stretches but does not shift the asymptote.

A cubic function has a graph that falls to the left and rises to the right, with two turning points. Determine the number of real x-intercepts it can have at most.

• 3
• 2
• 4
• 1

Answer: 3

A cubic can cross the x-axis at most 3 times; with two turning points it may cut the axis in three distinct places.

The number of bacteria is modelled by N(t) = 200(1.5)ᵗ, where t is in hours. Using technology, determine N(3).

• 675
• 900
• 450
• 300

Answer: 675

N(3) = 200 × 1.5³ = 200 × 3.375 = 675.

Using technology, find the x-coordinate of the local maximum of f(x) = −x³ + 3x, giving your answer to 3 s.f.

• 1.00
• −1.00
• 0.577
• 1.73

Answer: 1.00

f '(x) = −3x² + 3 = 0 → x² = 1 → x = ±1; the local maximum is at x = 1 (since the cubic has a negative leading coefficient).