Transformations of Graphs (DP IB Maths: AA SL)

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 .

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.