Graph transformations

The point P(6, –4) lies on the curve with equationy = f(x), x Find the point to which P is mapped when the curve with equationy = f(x) is transformed to the curve with equation

y = f(x + 2)

y = f^–1(x)

y = 2f(x) – 3

Figure 1 shows a sketch of a curve C with equationy = f(x), where f(x) is a quartic expression in x. The curve

• has maximum turning points at (–1, 0) and (5, 0)

• crosses the y ‑axis at (0, –75)

• has a minimum turning point at x = 2

Find the set of values of x for which

writing your answer in set notation.

Find the equation of C. You may leave your answer in factorised form. The curve C₁ has equation y = f(x) + k, where k is a constant. Given that the graph of C₁ intersects the x‑axis at exactly four places,

find the range of possible values for k.

The point P(3, –2) lies on the curve with equation y = f(x), x

Find the coordinates of the point to which P is mapped when the curve with equation y = f(x) is transformed to the curve with equation

y = f(x – 2)

y = f(2x)

y = 3f(–x) + 5

Figure 4 shows a sketch of the graph with equation

y = | 2 x – 3 k |

where k is a positive constant.

Sketch the graph with equation y = f (x) where

f (x) = k – | 2 x – 3 k |

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Figure 4 shows a sketch of the graph with equation y = | 2 x – 3 k | where k is a positive constant.

Sketch the graph with equation y = f (x) where f (x) = k – | 2 x – 3 k | stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

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

Show that f ʹ(x) = 8(2 + x − x²)e^−2x

Hence find, in simplest form, the exact coordinates of the stationary points of C. The function g and the function h are defined by g (x) = 2f (x) x h (x) = 2f (x) − 3 x 0

(c) Find (i) the range of g

the range of h

f(x) = 2x² + 4x + 9 x

Write f(x) in the form a(x + b)² + c, where a, b and c are integers to be found.

Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where

g(x) = 2(x – 2)² + 4x – 3 x

Find the range of the function

Figure 1 shows part of the graph with equation y = f (x), x . The graph consists of two line segments that meet at the point Q (6, –1). The graph crosses the y-axis at the point P (0, 11). Sketch, on separate diagrams, the graphs of

y = | f (x) |

y = 2f (– x) + 3

On each diagram, show the coordinates of the points corresponding to P and Q. Given that f (x) = a | x – b | – 1, where a and b are constants,

state the value of a and the value of b.

asymptote to the curve. On separate diagrams, sketch the curve with equation

(3) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the x-axis.

Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of

y = f(x),

y = | f(x) |,

y = –f(x – 4). Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote.

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

Find the value of ff (3).

On separate diagrams, sketch the curve with equation

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

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

y = 2f (3x)

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

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

y = 2f (x + 4),

y = f (x).

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

Figure 2 shows a sketch of the curve with the equation y = f (x), The curve has a turning point at A (3, 4) and also passes through the point (0, 5).

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

Sketch the curve with equation

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

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

Find f (x).

Explain why the function f does not have an inverse.

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

y = f(– x) + 1,

y = f(x + 2) + 3,

y = 2f(2 x). 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.

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

y = 2f (x) – 4,

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

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

The curve crosses the x- axis at (1, 0), touches it at (–3, 0) and crosses the y- axis at (0, –9)

In the space below, sketch the curve C with equation y = f(x + 2) and state the coordinates of the points where the curve C meets the x- axis.

Write down an equation of the curve C.

Use your answer to part (b) to find the coordinates of the point where the curve C meets the y-axis.

The curve C passes through the point (—1, 0) and touches the x-axis at the point (2, 0). The curve C has a maximum at the point (0, 4).

The equation of the curve C can be written in the form

where a, b andc are integers. Calculate the values of a, b and c. (5)

Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes. (3)

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

There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A.

Write down the coordinates of the point A.

On separate diagrams sketch the curve with equation

y = f(x + 3)

y = f(3 x) On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes.

The curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10).

Write down the value of k.

Figure 1 shows a sketch of the curve with equation y = f (x). The curve has a maximum point A at (–2, 3) and a minimum point B at (3, – 5). On separate diagrams sketch the curve with equation

y = f (x + 3)

y = 2f (x)

On each diagram show clearly the coordinates of the maximum and minimum points. The graph of y = f (x) + a has a minimum at (3, 0), where a is a constant.

Write down the value of a.

Figure 1 shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point (2, 5) and an asymptote y = 1, as shown in Figure 1. On separate diagrams, sketch the curve with equation

y = f(x) + 2

y = 4f(x)

y = f(x + 1)

On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.

Factorise completely x³ – 6x² + 9x

Sketch the curve with equation

showing the coordinates of the points at which the curve meets the x-axis.

Using your answer to part (b), or otherwise,

sketch, on a separate diagram, the curve with equation

showing the coordinates of the points at which the curve meets the x-axis.

Figure 1 shows a sketch of the curve C with equation y = f(x). There is a maximum at (0, 0), a minimum at (2, –1) and C passes through (3, 0). On separate diagrams sketch the curve with equation

y = f(x + 3),

y = f(– x).

On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the x-axis.

Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the point (0, 7) and has a minimum point at (7, 0). On separate diagrams, sketch the curve with equation

y = f(x) + 3,

y = f(2 x).

On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the y-axis.

Figure 1

Figure 1 shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the points (1, 0) and (4, 0). The maximum point on the curve is (2, 5). In separate diagrams sketch the curves with the following equations. On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the x-axis.

y = 2f(x),

y = f(– x).

The maximum point on the curve with equation y = f(x + a) is on the y-axis.

Write down the value of the constant a.