C3 Jan 2007

P2 January 2002The function f is defined by

f : x 2x – a, x ℝ,

where a is a positive constant.

(a) Sketch the graph of y = f(x), showing the coordinates of the points where the graph cuts the axes.

(2)(b) On a separate diagram, sketch the graph of y = f(2x), showing the coordinates of the points where the graph cuts the axes.

(2)

(c) Given that a solution of the equation f(x) = 12 x is x = 4, find the two possible values of a.

(4)

P2 January 2003(a) Sketch the graph of y = 2x + a, a > 0, showing the coordinates of the points where the

graph meets the coordinate axes. (2)

(b) On the same axes, sketch the graph of y =

1x .

(1)

(c) Explain how your graphs show that there is only one solution of the equation

x2x + a 1 = 0.(1)

(d) Find, using algebra, the value of x for which x2x + 1 1 = 0.(3)

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C3 Jan 2007

P2 January 2003Figure 3

y

(0, c)

O (d, 0) x

Figure 3 shows a sketch of the curve with equation y = f(x), x 0. The curve meets the coordinate axes at the points (0, c) and (d, 0).

In separate diagrams sketch the curve with equation

(a) y = f1(x), (2)

(b) y = 3f(2x).(3)

Indicate clearly on each sketch the coordinates, in terms of c or d, of any point where the curve meets the coordinate axes.

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C3 Jan 2007

P2 June 2003Figure 1

y

B(43 , 1)

1 O A(2, 0) 3 x

Figure 1 shows a sketch of the curve with equation y = f(x), 1 x 3. The curve touches the x-axis at the origin O, crosses the x-axis at the point A(2, 0) and has a maximum at the

point B(43 , 1).

In separate diagrams, show a sketch of the curve with equation

(a) y = f(x + 1), (3)

(b) y = f(x), (3)

(c) y = f(x), (4)

marking on each sketch the coordinates of points at which the curve

(i) has a turning point,

(ii) meets the x-axis.

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C3 Jan 2007

P2 January 2005Figure 1

Figure 1 shows a sketch of the curve with equation y = f(x), 0 x 4. The curve passes through the point (1, 0) on the x-axis and meets the y-axis at the point (0, –2).

Sketch, on separate axes, the graph of

(a) y =f(x),(2)

(b) y = f(2x),(2)

(c) y = f–1(x),(3)

in each case showing the coordinates of the points at which the graph meets the axes.

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(4, 0) (1, 0) x

(0, –2)

O

y

C3 Jan 2007

P2 June 2005Figure 1

Figure 1 shows part of the graph of y = f(x), x ℝ. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at (–1, 0) and the y-axis at (0, b), b < 0.

In separate diagrams, sketch the graph with equation

(a) y = f(x + 1), (2)

(b) y = f(x).(3)

Indicate clearly on each sketch the coordinates of any points of intersection with the axes.

Given that f(x) = x – 1 – 2, find

(c) the value of a and the value of b,(2)

(d) the value of x for which f(x) = 5x.(4)

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b

(1, a)

3 –1 x

y

O

C3 Jan 2007

P2 January 2006Figure 1

Figure 1 shows the graph of y = f(x), –5 x 5.

The point M (2, 4) is the maximum turning point of the graph.

Sketch, on separate diagrams, the graphs of

(a) y = f(x) + 3,(2)

(b) y = f(x),(2)

(c) y = f(x).(3)

Show on each graph the coordinates of any maximum turning points.

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–5

M (2, 4)

O 5 x

y

C3 Jan 2007

C3 June 2005Figure 1

Figure 1 shows part of the graph of y = f(x), x ℝ. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at (–1, 0) and the y-axis at (0, b), b < 0.

In separate diagrams, sketch the graph with equation(a)y = f(x + 1),

(2)(b)y = f(x).

(3)

Indicate clearly on each sketch the coordinates of any points of intersection with the axes.

Given that f(x) = x – 1 – 2, find(c) the value of a and the value of b,

(2)(d) the value of x for which f(x) = 5x

(4)

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b

(1, a)

3 –1 x

y

O

C3 Jan 2007

C3 January 2006

Figure 1

Figure 1 shows the graph of y = f(x), –5 x 5.The point M (2, 4) is the maximum turning point of the graph.Sketch, on separate diagrams, the graphs of

(a) y = f(x) + 3,(2)

(b) y = f(x),(2)

(c) y = f(x).(3)

Show on each graph the coordinates of any maximum turning points.

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–5

M (2, 4)

O 5 x

y

C3 Jan 2007

C3 June 2006Figure 1

Figure 1 shows part of the curve with equation y = f(x), x ℝ, where f is an increasing function of x. The curve passes through the points P(0, –2) and Q(3, 0) as shown.

In separate diagrams, sketch the curve with equation

(a) y = f(x),(3)

(b) y = f–1(x),(3)

(c) y = 12 f(3x).

(3)

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

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(3, 0)

P(0, –2)

O

y

x

y = f(x)

Q

C3 Jan 2007

C3 January 2007f(x) = x4 – 4x – 8.

(d) Sketch the graph of y = f(x).(3)

(e) Hence sketch the graph of y = f(x).(1)

C3 January 2008

Figure 1

Figure 1 shows a sketch of the curve with equation y = f (x).

The curve passes through the origin O and the points A(5, 4) and B(–5, –4).

In separate diagrams, sketch the graph with equation

(a) y =f (x),(3)

(b) y = f (x) ,(3)

(c) y = 2f(x + 1) .(4)

On each sketch, show the coordinates of the points corresponding to A and B.

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C3 Jan 2007

C3 June 2008

Figure 1

Figure 1 shows the graph of y = f(x), x ℝ,

The graph consists of two line segments that meet at the point P.

The graph cuts the y-axis at the point Q and the x-axis at the points (–3, 0) and R.

Sketch, on separate diagrams, the graphs of

(a) y = f(x),(2)

(b) y = f (– x).(2)

Given that f(x) = 2 – x + 1,

(c) find the coordinates of the points P, Q and R,(3)

(d) solve f(x) = 12 x.

(5)

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C3 Jan 2007

C3 January 2009

Figure 1

Figure 1 shows the graph of y = f (x), 1 < x < 9.

The points T(3, 5) and S(7, 2) are turning points on the graph.

Sketch, on separate diagrams, the graphs of

(a) y = 2f(x) – 4,(3)

(b) y = f(x).(3)

Indicate on each diagram the coordinates of any turning points on your sketch.

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C3 Jan 2007

C3 June 2009

Figure 2

Figure 2 shows a sketch of part of the curve with equation y = f(x), x ℝ.

The curve meets the coordinate axes at the points A(0, 1 – k) and B(12 ln k, 0), where k is a

constant and k >1, as shown in Figure 2.

On separate diagrams, sketch the curve with equation

(a) y = f(x), (3)

(b) y = f −1(x).(2)

Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.

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C3 Jan 2007

C3 January 2010

Figure 1

Figure 1 shows a sketch of the graph of y = f (x).

The graph intersects the y-axis at the point (0, 1) and the point A(2, 3) is the maximum turning point.

Sketch, on separate axes, the graphs of

(i) y = f(–x) + 1,

(ii) y = f(x + 2) + 3,

(iii) y = 2f(2x) .

On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the coordinates of the point to which A is transformed.

(9)

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C3 Jan 2007

C3 June 2010

Figure 2

Figure 2 shows a sketch of the curve with the equation y = f(x), x ℝ.

The curve has a turning point at A(3, − 4) and also passes through the point (0, 5).

(a) Write down the coordinates of the point to which A is transformed on the curve with equation

(i) y = |f(x)|,

(ii) y = 2f(12 x).

(4)(b) Sketch the curve with equation y = f(|x|).

On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis.

(3)

The curve with equation y = f(x) is a translation of the curve with equation y = x2.

(c) Find f(x).(2)

(d) Explain why the function f does not have an inverse.(1)

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C3 Jan 2007

C3 June 2011

Figure 1

Figure 1 shows part of the graph of y = f (x), x ℝ.

The graph consists of two line segments that meet at the point R (4, – 3), as shown in Figure 1.

Sketch, on separate diagrams, the graphs of

(a) y = 2f(x + 4),(3)

(b) y = f(−x) .(3)

On each diagram, show the coordinates of the point corresponding to R.

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C3 Jan 2007

C3 January 2012

Figure 1

Figure 1 shows the graph of equation y = f(x).

The points P (– 3, 0) and Q (2, – 4) are stationary points on the graph.

Sketch, on separate diagrams, the graphs of

(a) y = 3f(x + 2),(3)

(b) y = f(x).(3)

On each diagram, show the coordinates of any stationary points.

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C3 Jan 2007

C3 June 2012

Figure 2

Figure 2 shows part of the curve with equation y = f(x).The curve passes through the points P(−1.5, 0)and Q(0, 5) as shown.

On separate diagrams, sketch the curve with equation

(a) y = f(x)(2)

(b) y = f(x)(2)

(c) y = 2f(3x)(3)

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

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C3 Jan 2007

C3 January 2013

Figure 1

Figure 1 shows part of the curve with equation y = f(x) , x ℝ.

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

On separate diagrams, sketch the curve with equation

(b) y = f −1(x),(2)

(c) y = f(x) − 2,(2)

(d) y = 2f(12 x ) .

(3)

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

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C3 Jan 2007

ANSWERS

C3 June 2005

(c) a = –2, b = –1 (d) x = –

16

C3 January 2006 N/A

C3 June 2006N/A

C3 January 2007N/A

C3 January 2008(a) (5, 4), (–5, 4) (b) (5, 4), (–5, 4) (c) (4, 8), (–6, –8)

C3 June 2008(c) P(–1, 2), Q (0, 1), R(1, 0) (d)

C3 January 2009N/A

C3 June 2009N/A

C3 January 2010N/A

C3 June 2010(a) (i) (3, 4) (ii) (6, –8) (c) f(x) = (x – 3)2 – 4

C3 June 2011(a) R = (0, –6) (b) R = (–4, 3)

C3 January 2012(a) (–5, 0), (0, –12) (b) (–3, 0), (2, 4)

C3 June 2012(a) (–1.5, 0) and (0, 5) (b) (0, 5) (c) (0, 10) and (–0.5, 0)

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