2.5 Transformations of Graphs

The point  lies on the curve with equation .  

State the coordinates of the image of point  on the curves with the following equations:

The point  lies on the curve with equation

On the graph of , where is a constant, the point  is mapped to the point .  Determine the value of  .

On the graph of , where is a constant, the point  is mapped to the point   Determine the value of .

On the graph of , where is a constant, the point is mapped to the point .  Determine the value of .

On the graph of , where is a constant, the point is mapped to the point .  Determine the value of .

The diagram below shows the graph of The two marked points  and  lie on the graph.

In separate diagrams, sketch the curves with equation

On each diagram, give the coordinates of the images of points and  under the given transformation.

On the graph of  the image of one of the two marked points has an  coordinate of 2.  Find the two possible values of .

The diagram below shows the graph of .  The marked point  lies on the graph, and the graph meets the origin at the marked point .

In separate diagrams, sketch the curves with equation

On each diagram, give the coordinates of the images of points  and  under the given transformation.

On the graph of  the image of one of the two marked points has a coordinate of 4. Find the value of .

The diagram below shows the graph of .  The graph intersects the coordinate axes at the two marked points  and .  The graph has two asymptotes as shown, with equations     and 

In separate diagrams, sketch the curves with equation

(i)         

(ii)       

On each diagram, give the coordinates of the images of points  and  under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of  has an asymptote at one of the coordinate axes.  Find the value of .

Describe, in order, a sequence of transformations that maps the graph of  onto the following graphs:

Given that  find an expression for , where  is obtained by applying the following sequence of transformations to .

Translation by 

Vertical stretch of scale factor 4

Translation by 

Find an expression for .

 Find an expression for .

What can you deduce about the sequence of transformations given
by ?

The equation , where , with , is shown below.

The points and  are the points where the graph intercepts the coordinate axes.

Write down, in terms of , the coordinates of and .

Sketch the graph of , labelling the images of the points and stating their coordinates in terms of .

Write down the value of a such that the point  is three times as far from the origin as the point .

The function  is to be transformed by a sequence of functions, in the order detailed below:

A horizontal stretch by scale factor 2

A reflection in the -axis

A translation by 

Write down an expression for the combined transformation in terms of .

The diagram shows the graph of , where ,

Write down the maximum value of  when .

Write down the first value of  for which this maximum occurs.

Write down the minimum value of  when .

Write down the first value of  for which this minimum occurs.

Find, in terms of , the combination of transformations that would map the graph of  onto the graph of ,  .

Let .

Write down the value of .

The function f can be written in the form of.

Find the values of a, h and k.

The graph of g is obtained from the graph of f by a reflection in the x-axis followed by a translation by the vector .

Find, giving your answer in the form of

Let f and g be functions defined for such that 

The graph of is obtained from the graph of f after the follow transformations:

a horizontal stretch by a factor of v,

followed by

a translation by the vector .

Write down the values of

The point  A (2 , 0)  on the graph of is mapped to point B on the graph of g.

Find the distance between points A and B, giving your answer in the form where p and q are integers to be found, and where q has no square number factors.

Let   and

Show that the graph of is a translation of the graph of f, and find the vector that translates the graph of f onto the graph of g.

The diagram below shows parts of the graphs of f and g.  Point A is the y-intercept of f, point B is the intersection between the graphs of f and g and point C is the y-intercept of g.

Find the area of the triangle that has points A, B and C as its vertices.

Let  for .

Sketch the graph of  on the grid below, clearly labelling any intersections the graph makes with the coordinate axes.

The graph of f is reflected in the x-axis and then translated by the vector  to obtain the graph of   .

Show that the equation   has no solutions.

The function f is defined by

The graph of the function g is obtained by applying the following transformations to the graph of f:

a reflection in the y-axis.

followed by 

a translation by the vector ,

Find .

Let , where .

The graph of a function g is obtained when the graph of f is transformed by

 a reflection in the y-axis,

 followed by

 a vertical stretch by a factor of   .

Find , giving your answer in the form   .

A particle moves along a straight line so that its velocity in m s^-1 at time x seconds is given by

Find the value of x when the particle’s velocity is 85 m s^-1.

Let   where .

The equation  has two equal roots.

Given that , find the values of a and c.

Find the coordinates of the vertex of the graph of f.

The graph of a function g is obtained from the graph of f  by a reflection in the y -axis, followed by a horizontal stretch by a factor of .

Find an expression for , along with the coordinates of the y -intercept of the graph of g.

Using the geometric nature of the two transformations by which the graph of g was obtained from the graph of f, explain why the graphs of f and g must have the same  y -intercept.

Let  where The graph of f has x- intercepts at  and , with  lying on the negative x-axis and  lying on the positive x-axis.  The y-intercept of the graph is at   and the vertex of the graph lies in the fourth quadrant.  This information is represented on the diagram below.

The graph of a function g is obtained from the graph of  f  by a translation by the vector ,  followed by a reflection in the y-axis.  Point A on the graph of f has an x-coordinate of 1 and is mapped to point B on the graph of g.

Find the coordinates of B.