Hint
Use the stated units and precision. Complete the square, then use the vertex form and its positive leading coefficient.
Worked solution
(a)
\(f(x)=2 \left(x - 4\right)^{2} + 2\) expands to the given expression. The square vanishes at \(\displaystyle x=4\).
Answer: \(2 \left(x - 4\right)^{2} + 2\); vertex \(\left( 4, \ 2\right)\)
(b)
The square is nonnegative and its coefficient 2 is positive. Equality is attained at \(\displaystyle x=4\).
Answer: \(f(x)\ge 2\)
(c)
\((x-4)^2=4\) gives both signs: \(\displaystyle x=4\pm 2\).
Answer: \(\left\{2, 6\right\}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 3 marks
- 1 mark: Factor the leading coefficient and complete the square, including the correction.
- 1 mark: Read both vertex coordinates from the resulting form.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 2 marks
- 1 mark: Use nonnegativity of the square and identify where equality occurs.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the answer against the original conditions.
(c) · 2 marks
- 1 mark: Isolate the square and retain both real square-root branches.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
Total: 7 marks.