Modulus Functions (CIE A Level Maths: Pure 3)

On the same axes, sketch the graphs of    and    where

Label the points at which the graphs intersect the coordinate axes.

Solve the equation .

Solve the equation  

The functions  are defined as follows

Sketch the graph of , stating the coordinates of all points where the graph intercepts the coordinate axes.

Write down the solutions to the equation 

The turning point on the graph of has coordinates  as shown on the diagram below.

On the diagram above sketch the graph of  and state the coordinates of the turning point.

State the distance between the turning points on the graphs of and .

The diagram below shows the graph of  where

Write down the equations of the two asymptotes.

Determine the equations of the two asymptotes on the graph of .

Determine the range of .

On the same axes, sketch the graphs of  where

Label the points at which the graphs intersect the coordinate axes.

Solve the equation 

Which of the solutions to    is also a solution to  ?

The function    is defined as

Explain why the inverse of    does not exist.

Suggest an adaption to the domain of  so its inverse does exist, but also produces the maximum possible range for .

Using your adaption from part (b), find an expression for   and state its domain and range.

Solve the equation  , giving your answers in exact form.

The functions are defined as follows

Sketch the graph of  , stating the coordinates of all points where the graph intercepts the coordinate axes.

Solve the equation  

The minimum point on the graph of has coordinates as shown on the diagram below.

Sketch the graph of  and state the coordinates of the maximum point.

Find the exact distance between the minimum point on the graph of and the maximum point on the graph of 

On the same axes sketch the graphs of and , where .

Find an expression for  and state its domain.

Show that 

The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius 10 m, as shown in the diagram below.

At time seconds, the -coordinate of the boat is modelled by the function

and the height,  m, of the boat above the ground, at time seconds, is modelled by

Verify that the initial position of the boat is .

Find the position of the boat when it has swung through an angle of   anticlockwise from the -axis, as shown in the diagram above. 
Find the time at which the boat first reaches this position.

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

Solve the equation  , giving your answers in exact form.

The functions    are defined as follows

Sketch the graph of  , stating the coordinates of all points where the graph intercepts the coordinate axes.

There are between 0 and 4 solutions to the equation  , where is a real number.  Determine the values of that produce each number of solutions.

On the same axes, sketch the graphs of  and  where

Label the points at which the graphs intersect the coordinate axes.

Solve the equation  

Which of the solutions to  is not a solution to  ?

A sketch of the graph with equation where  is shown below.
Points and are the -axis intercepts and point  is the maximum point on the graph.

On the diagram above, sketch the graph of labelling the image of the points and with .

Show that the area of triangle is twice the area of triangle .

The function  is transformed by a sequence of transformations as described below.

1. Horizontal stretch by scale factor 3,
2. The modulus of the function is then taken,
3. Reflection in the -axis.

Write down the resulting transformation in terms of as well as an expression in terms of .

A swing boat fairground ride is modelled as moving forwards and backwards along the path of a semi-circle, radius 18 m, as shown in the diagram below.

Show that, for 

the -coordinate of the boat is given by   ,

the -coordinate is given by  

The model is refined so that the coordinates of the boat can be calculated from the time, seconds, after the boat is set in motion.   The and  coordinates are now given by 

where  is a constant.

Briefly explain why the modulus of  is required for the - coordinate.

Given that the time between the boat reaching its maximum height at either end of the ride is 8 seconds, find the value of .

For , find the times when the boat is equidistant from the ground and horizontally from the origin.