4400 Higher June 2009 Paper 3 Q15

EdexcelHigherOld spec8 marksDrawing & Using Graphs Functions

15 The diagram shows part of the graph of \(y = \mathrm{f}(x)\) and part of the graph of \(y = \mathrm{g}(x)\).

Graphs of the curve y = f(x) and the straight line y = g(x) for x from 0 to 7 and y from 0 to 25, with a cross at (1, 16) on the curve

(a) Find \(\mathrm{f}(3)\). (1)

(b) Solve \(\mathrm{f}(x) = \mathrm{g}(x)\).
Give your answers correct to 1 decimal place. (2)

(d) Find an estimate for the gradient of the graph of \(y = \mathrm{f}(x)\) at the point (1, 16). (3)

Scheme	Marks
12	B1
	(1)

Mark scheme (b)

Scheme	Marks
0.2 3.6 6.1 or 6.2 or values rounding to these	B2
	(2)

Notes

B2: for all 3 correct solutions
(B1 for 2 correct solutions or for 3 coordinates with correct solutions as \(x\)-coordinates)

Scheme	Marks
5 seen	M1
0	A1
	(2)

Mark scheme (d)

Scheme	Marks
tan drawn at (1, 16)	M1
\(\dfrac{\text{vertical difference}}{\text{horizontal difference}}\)	M1
6 − 10 inc	A1
	(3)
	(8 marks)

Notes

M1: tan or tan produced passes between points (0.5, \(11 \leqslant y \leqslant 13\)) and (1.5, \(19 \leqslant y \leqslant 21\))

M1: finds their \(\dfrac{\text{vertical difference}}{\text{horizontal difference}}\) for two points on tan

or finds the intercept of their tangent on the \(y\)-axis and substitutes \(y = 16\), \(x = 1\) and their \(c\) into \(y = mx + c\)

or finds their \(\dfrac{\text{vertical difference}}{\text{horizontal difference}}\) for two points on curve, where one of the points has an \(x\)-coordinate between 0.5 and 1 inc and the other point has an \(x\)-coordinate between 1 and 1.5 inc

A1: dep on both M marks