O-LevelAdditional MathematicsFunctions4037/22 · May/June 2011 · Q1011 Marks
A function f is defined by f : x → 18 + 16x – 2x² for x ∈ R.
- (a)(i)Express 18 + 16x – 2x² in the form a + b(x + c)², where a, b and c are integers.[3]
- (a)(ii)Write down the coordinates of the stationary point on the graph of y = f(x).[1]
- (a)(iii)Sketch the graph of y = f(x).[2]
- (b)(i)A function g is defined by g : x → (x + 3)² – 7 for x > −3. Find an expression for g⁻¹(x).[2]
- (b)(ii)Solve the equation g⁻¹(x) = g(0).[3]
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Mark scheme
Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.
- [1] (a)(i) a = 50
- [1] (a)(i) b = -2 OR 50-2(x-4)²
- [1] (a)(i) c = -4
- [1] (a)(ii) (4, 50)
- [1] (a)(iii) Correct shape (inverted parabola)
- [1] (a)(iii) Maximum and y intercept in correct quadrant (Maximum at (4,50), y-intercept at (0,18))
- [1] (b)(i) Method for inverse (e.g., y=(x+3)²-7 => x+3=√(y+7) => x=√(y+7)-3)
- [1] (b)(i) √(x+7)-3
- [1] (b)(ii) g(0) = 2
- [1] (b)(ii) Solves g⁻¹(x) = 2 or solves x = g(g(0)) = g(2)
- [1] (b)(ii) 18
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