Transformations of Graphs (DP IB Maths: AI HL)

Let and , for .

Give a full geometric description of the two individual transformations that can be combined to obtain the graph of f from the graph of g.

The graph of f is translated by the vector to give the graph of h.

Now consider the graph of h as a transformation of the graph of g.  The point A on the graph of h corresponds to the point (2, 8) on the graph of g.

Find the coordinates of A.

Let f and g be functions such that ,  for .

The transformation that maps the graph of f onto the graph of g may be represented as a combination of two simpler transformations:

a vertical stretch by a factor of v, 

followed by

a translation by the vector .

Write down the values of

The point A(3, 4) on the graph of f is mapped to point B on the graph of g.

Find the coordinates of B.

Let for .

Sketch the graph of  on the following grid in the interval  .  Use an appropriate scale and clearly label any intersections the graph makes with the coordinate axes.

Find

The function g is obtained when the graph of f is translated by the vector .

Sketch the graph of on the same grid above, also for the interval .  Clearly label any intersections the graph makes with the coordinate axes and label the graph in the form   where a, b and c are constants to be determined.

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 .

Find an expression for .

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 translation by the vector ,

followed by 

a reflection in the x-axis.

Find an expression for .

The following diagram shows the graph of

Write down the value of 

Find the value of .

Given that , find the domain and range of g.

Let ,  where .

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

a vertical stretch by a factor of  , 

followed by

a translation by the vector.

Find , giving your answer in the form

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

Find the value of t when the particle’s velocity is 11 m s^-1.

Let ,  for .

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

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 with scale factor  .

Find an expression for giving your answer in the form 

Let,  for , where .

Given that the equation   has two equal roots, and that  ,

find the value of b.

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 with scale factor 2.

Find an expression for and state the coordinates of the y-intercept of the graph of g.

Let

For the graph of f, find

The graph of a function 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 .

Consider the functions  and defined by  and , where each function has the largest possible valid domain.

Write down the domain of .

The graph of  can be transformed onto the graph of  by a single translation and a single stretch, both of which are parallel to one of the coordinate axes.

Describe the sequence of transformations in the case where:

The graph of  can be also transformed onto the graph of  by a single translation using the vector .

Find the exact values of  and .

The graph of a function  is shown below. The points A  and B lie on the graph and are a local maximum and a local minimum respectively.  The -axis is an asymptote to the graph.

On separate sets of axes, sketch the graphs of:

.

.

In each case give the coordinates of the points onto which A and B are mapped, and state the equation of the asymptote.

The graph of  is stretched horizontally by a scale factor of  then translated by the vector  to map it onto the graph of  .

Find the values of  and .

Consider the function  defined by 

Find  the coordinates of

the -intercept

the -intercept

of the graph of  .

Sketch the graph of .

The graph of  is first reflected in the -axis and then translated by the vector  to obtain the graph of a function .

Find an expression for .

Let  and  for

Give a full geometric description of two individual transformations that can be combined to obtain the graph of f  from the graph of g, given that:

(i)     a stretch is to be applied first, followed by a translation

(ii)    a translation is to be applied first, followed by a stretch.

The graph of f is translated by the vector to give the graph of h.

Now consider h as a transformation of g. The point where   is translated to point A on the graph of h.

Find the coordinates of A.

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.

Consider the function  defined by .  

The graph of  is stretched horizontally by scale factor of 0.75 and then translated by the vector .  The function corresponding to the transformed graph is denoted .

Write down an expression for the function .

Consider the function defined by 

The graph of  can be transformed onto the graph of  by a horizontal stretch and a vertical stretch.

Describe the stretches needed to map the graph of  onto the graph of .

Let  be a function.

For each of the functions below, describe two different sequences of transformations that map the graph of  onto the graph of the function.  Each sequence should consist of exactly two basic transformations (translations, stretches, or reflections).

.

The graph of the function  defined by  is shown below. There is a local minimum at A  and a local maximum at B.

Sketch the graph of . Clearly state the coordinates of the points corresponding to the points A and B.