IAL January 2019

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8.

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(4, 9)

(0, 6) y = f (x)

y =4

(−3, 0) O x

Figure 4

The curve C with equation y = f ( x ) is shown in Figure 4.

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The curve C
• has a single turning point, a maximum at (4, 9)
• crosses the coordinate axes at only two places, (−3, 0) and (0, 6)
• has a single asymptote with equation y = 4
as shown in Figure 4.

(a) State the equation of the asymptote to the curve with equation y = f (− x ).
(1)

1 
(b) State the coordinates of the turning point on the curve with equation y = f  x .
4 
(1)

Given that the line with equation y = k, where k is a constant, intersects C at

exactly one point,
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(c) state the possible values for k. DO NOT WRITE IN THIS AREA
(2)

The curve C is transformed to a new curve that passes through the origin.

(d) (i) Given that the new curve has equation y = f ( x ) − a , state the value
of the constant a.

(ii) Write down an equation for another single transformation of C that

also passes through the origin.
(2)

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 IAL June 2019
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10. A curve has equation y = f(x), where
2
f(x) = (xí x

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The curve touches the x-axis at the point P and crosses the x-axis at the point Q.

(a) State the coordinates of the point P.

(1)

(b) Find I‫ މ‬x).

(4)

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(c) Hence show that the equation of the tangent to the curve at the point where x =
2
can be expressed in the form y = k, where k is a constant to be found.
(3)

The curve with equation y = f(x + a), where a is a constant, passes through the origin O.

(d) State the possible values of a.

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(2)
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 IAL January 2020
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10. The curve C1 has equation y = f(x), where
2
f(x) = (4x – 3)(x – 5)

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(a) Sketch C1 showing the coordinates of any point where the curve touches or crosses
the coordinate axes.
(3)

(b) Hence or otherwise

⎛1 ⎞
(i) find the values of x for which f ⎜ x⎟ = 0
⎝4 ⎠
(ii) find the value of the constant p such that the curve with equation y = f(x) + p
passes through the origin.
(2)

A second curve C2 has equation y = g(x), where g(x) = f(x + 1)

(c) (i) Find, in simplest form, g(x). You may leave your answer in a factorised form.

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(ii) Hence, or otherwise, find the y intercept of curve C2
(3)

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 IAL October 2020
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5. (i) y

–5 O 2 x
–3

y = f(x)

Figure 2

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

The curve passes through the points (–5, 0) and (0, –3) and touches the x‑axis at the
point (2, 0).

On separate diagrams sketch the curve with equation

(a) y = f(x + 2)

(b) y = f(–x)

On each diagram, show clearly the coordinates of all the points where the curve cuts
or touches the coordinate axes.
(6)
(ii) y

O p q x

Figure 3

Figure 3 shows a sketch of the curve with equation

π
y = k cos  x +  0  x  2π
 6
where k is a constant.

The curve meets the y‑axis at the point (0, 3 ) and passes through the points ( p, 0)
and (q, 0).

Find

(a) the value of k,

(b) the exact value of p and the exact value of q.

(3)

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 IAL October 2021
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9. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

y = f(x)
P(9, 3)

O x

Figure 5

Figure 5 shows a sketch of the curve with equation y = f(x) where

f(x)   = x          x > 0

The point P(9, 3) lies on the curve and is shown in Figure 5.

On the next page there is a copy of Figure 5 called Diagram 1.
(a) On Diagram 1, sketch and clearly label the graphs of

y = f(2x) and y = f(x)  + 3

Show on each graph the coordinates of the point to which P is transformed.

(3)

The graph of y = f(2x) meets the graph of y = f(x) + 3 at the point Q.

(b) Show that the x coordinate of Q is the solution of

x =3 ( 2 +1 )
(3)
(c) Hence find, in simplest form, the coordinates of Q.
(3)
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 IAL October 2021
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Question 9 continued
y

y = f(x)
P(9, 3)

O x

Diagram 1

Turn over for a copy of Diagram 1 if you need to redraw your graphs.

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 IAL January 2022
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7.
y

O x

Figure 3

Figure 3 shows a sketch of part of the curve with equation y = f(x), where

f(x) = (x + 4)(x – 2)(2x – 9)

Given that the curve with equation y = f(x) – p passes through the point with
coordinates (0, 50)

(a) find the value of the constant p.

(2)

Given that the curve with equation y = f(x + q) passes through the origin,

(b) write down the possible values of the constant q.

(2)

(c) Find f ′(x).

(4)

(d) Hence find the range of values of x for which the gradient of the curve with equation
y = f(x) is less than –18
(3)
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 IAL June 2022
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4.
y Q

y = f(x)
y=1

P O x

Figure 1

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

3 
The curve has a minimum at P(−1, 0) and a maximum at Q  , 2
2 

The line with equation y = 1 is the only asymptote to the curve.

On separate diagrams sketch the curves with equation

(i) y = f(x) − 2
(3)

(ii) y = f(−x)
(3)

On each sketch you must clearly state

• the coordinates of the maximum and minimum points
• the equation of the asymptote

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 IAL October 2022

7.
y

(–4, 6) (–1, 6) (2, 6) (3, 6)

O x

y = f(x)

Figure 1

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

The points P(–4, 6), Q(–1, 6), R(2, 6) and S(3, 6) lie on the curve.
(a) Using Figure 1, find the range of values of x for which

f(x) < 6
(3)
(b) State the largest solution of the equation

f(2x) = 6
(1)

(c) (i) Sketch the curve with equation y = f(–x).

On your sketch, state the coordinates of the points to which P, Q, R and S
are transformed.
(ii) Hence find the set of values of x for which

f(–x)  6 and x < 0

(4)
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 IAL January 2023

7. (a) On Diagram 1, sketch a graph of the curve C with equation

6
y= x≠0
x
(2)
6
The curve C is transformed onto the curve with equation y = x≠2
x−2
(b) Fully describe this transformation.
(2)
The curve with equation

6
y= x≠2
x−2

and the line with equation

y = kx + 7    where k is a constant

intersect at exactly two points, P and Q.

Given that the x coordinate of point P is –4
(c) find the value of k,
(2)
(d) find, using algebra, the coordinates of point Q.

(Solutions relying entirely on calculator technology are not acceptable.)

(4)

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 IAL January 2023

Question 7 continued
y

O x

Diagram 1

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Turn over for a copy of Diagram 1 if you need to redraw your graph.

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