Graphs of Functions (OCR A Level Maths: Pure)

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.

Sketch the graph of  and write down the equations of any asymptotes.

On the axes below sketch the graphs of both  and .

Using your graph, or otherwise, find the solution to the simultaneous equations

and
.

 is inversely proportional to .  When .  Find the constant of proportionality and sketch the graph of  against .

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.

On the same diagram, sketch the graphs of  and .

Use your graph to determine the number of solutions to the equation

.

A machine computes a calculation in time,  seconds, that is proportional to the number of processes, , involved.  For a calculation involving 10 processes the machine takes 0.01 seconds.

Show that the constant of proportionality is 0.001.

Hence write down an equation linking the number of processes  to the time taken .

Find the time it takes for the machine to compute a calculation involving 200 processes.

How many processes are involved for a calculation taking 2.3 seconds?

The diagram below shows the graph of  , where .

Sketch the graph of , where .

State which graph the following points must lie on and find the value of in each case.

 

Show that  is a factor of .

Fully factorise .

Sketch the graph of .
Label any points where the graph crosses the coordinate axes.

Solve the equation .

Write down the -coordinates of the points of intersection between the graphs of  and .

Find the -coordinates of the points of intersection between the graphs of  and .

On the same diagram, sketch the graphs of  and .

Given that  is a factor of , sketch the graph of . Label any points where the graph intersects the coordinate axes.

(There is no need to label any stationary points.)

Sketch the graph of  and write down the equations of any asymptotes.

On the axes below sketch the graphs of both  and .

Using your graph, or otherwise, find the solutions to the simultaneous equations

 is inversely proportional to .  When .  Find the constant of proportionality and sketch the graph of  against .

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.

On the same diagram, sketch the graphs of  and .

Use your graph to determine the number of solutions to the equation

.

A machine computes a calculation in time,  seconds, that is proportional to the square of the number of processes, , involved.  For a calculation involving 8 processes the machine takes 0.032 seconds.

Find an equation linking the number of processes, () to the time taken, () seconds.

How many processes are involved for a calculation taking 0.2 seconds?

Find the time it takes for the machine to compute a calculation involving 30 processes.

The diagram below shows the graph of the equation , where .

Sketch the graph of , where .

Given that  is a negative real number, state with a reason, which graph passes through the point .

Show that  is a factor of the function  and hence, or otherwise, fully factorise .

Sketch the graph of .
Label any points where the graph crosses the coordinate axes.

Find the coordinates of the points of intersection between the curve with equation  and the line with equation .

On the same diagram, sketch the graphs of  and .
Label the coordinates of any points of intersection between the two graphs.
Also label any points where the graphs intersect the coordinate axes.

 are factors of , where .

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.  (There is no need to label any stationary points).

On the same diagram, sketch the graphs of  and .

Write down the equation(s) of any lines of symmetry and asymptotes for the two graphs in part (a).

On the axes below sketch the graphs of both  and .

Using your graph, or otherwise, find the solutions to the equation

.

 is inversely proportional to the square of .  When .  Find the constant of proportionality and sketch the graph of  against .

Sketch the graph of  labelling any points where the graph intersects the coordinate axes.

On the same diagram, sketch the graphs of  and .

Write down the number of solutions to the equation

.

A machine computes a calculation in time,  seconds, that is proportional to the cube root of the number of processes, , involved.  For a calculation involving 8 processes the computer takes  seconds.

How many processes are involved for a calculation taking  seconds?

Find the time it takes for the machine to compute a calculation involving 250 processes.

On separate diagrams, sketch the graphs of , where  and , where .

One of the graphs passes through the point with coordinates .

Write  in terms of  and, justifying your answer, state which graph this point must lie on.

Fully factorise .

Sketch the graph of .
Label any points where the graph crosses the coordinate axes.

On the same diagram, sketch the graphs of and .

Label the coordinates of any points of intersection between the two graphs.
Also label any points where the graphs intersect the coordinate axes.