C3 January 2013 Q3

EdexcelOld spec9 marksFunctions (including |mod|)Sketching & Transforming Graphs

Figure 1: increasing curve y = f(x) through P(−3, 0) and Q(0, 2) — Figure 1

Figure 1 shows part of the curve with equation \(y = \mathrm{f}(x),\ x \in \mathbb{R}\).

The curve passes through the points \(Q(0, 2)\) and \(P(-3, 0)\) as shown.

(a) Find the value of \(\mathrm{ff}(-3)\). (2)

On separate diagrams, sketch the curve with equation

(b) \(y = \mathrm{f}^{-1}(x)\), (2)

(d) \(y = 2\mathrm{f}\left(\tfrac{1}{2}x\right)\). (3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

Mark scheme (a)

Scheme	Marks
ff(-3)= f(0),=2	M1,A1
	(2)

Notes

M1 A full method of finding ff(-3). f(0) is acceptable but f(-3)=0 is not.
Accept a solution obtained from two substitutions into the equation \(y = \dfrac{2}{3}x + 2\) as the line passes through both points. Do not allow for \(y = \ln(x + 4)\), which only passes through one of the points.

A1 Cao ff(-3)=2. Writing down 2 on its own is enough for both marks provided no incorrect working is seen.

Sketch: y = f⁻¹(x), increasing curve through (0, −3) and (2, 0) — Mark scheme (b)

Sketch: y = f(|x|) − 2, symmetrical about the y-axis with a cusp at (0, 0) — Mark scheme (c)

Sketch: y = 2f(x/2), increasing curve through (−6, 0) and (0, 4) — Mark scheme (d)

Scheme	Marks
Shape	B1
(0,-3) and (2,0)	B1
	(2)

Scheme	Marks
Shape	B1
(0,0)	B1
	(2)

Scheme	Marks
Shape	B1
(-6,0) or (0,4)	B1
(-6,0) and (0,4)	B1
	(3)
	(9 marks)