Hint
The leading coefficient is nonzero. Complete the square.
Worked solution
(a)
\(3 \left(x - 9\right)^{2} - 12 = 3 \left(x - 9\right)^{2} - 12\); the minimum is -12 at \(\displaystyle x=9\).
Answer: \(3 \left(x - 9\right)^{2} - 12\); minimum \(-12\) at \(\displaystyle x=9\).
(b)
\(\left(x - 9\right)^{2} = 4\) gives the two square-root branches.
Answer: \(\left\{7, 11\right\}\)
(c)
The upward-opening graph lies on or below the axis between its roots.
Answer: \(\left[7, 11\right]\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Complete the square and use the positive leading coefficient to locate the minimum.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The vertex x-coordinate is h, not −h.
(b) · 2 marks
- 1 mark: Set the completed square equal to \(\displaystyle d^{2}\) and retain both real roots.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A discriminant counts roots; it does not give their coordinates.
(c) · 2 marks
- 1 mark: Use the two roots and upward opening to choose the closed inner interval.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The vertex x-coordinate is h, not −h.
Total: 6 marks.