Hint
Determine the sign of the original quadratic first. Set the original quadratic to zero.
Worked solution
(a)
The original expression is negative between 11 and 15; reflect precisely that part. The minima are (11,3) and (15,3).
Answer: \(\begin{cases} 7 - \left(x - 13\right)^{2} & \text{for}\: x \geq 11 \wedge x \leq 15 \\\left(x - 13\right)^{2} - 1 & \text{otherwise} \end{cases}\); sharp minima \(\left( 11, \ 3\right)\) and \(\left( 15, \ 3\right)\).
(b)
The centre is reflected upwards, while the outer branches are unbounded.
Answer: \(\left( 13, \ 7\right)\); \(\left[3, \infty\right)\)
(c)
Reflect the negative vertex value.
\(7\)
Answer: \(7\)
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) · 3 marks
- 1 mark: Determine where the underlying quadratic is negative and reflect exactly that portion.
- 1 mark: Give both branch rules and locate both sharp minima.
- 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: Do not reflect the positive outer branches.
(b) · 2 marks
- 1 mark: Evaluate the reflected vertex and use the unbounded outer branches to determine the range.
- 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 sharp minimum need not have a derivative.
(c) · 2 marks
- 1 mark: Reflect the negative vertex value.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not reflect the positive outer branches.
Total: 7 marks.