Graphs of Functions (Edexcel A Level Maths: Pure): Exam Questions

Sketch the curve with equation

where is a positive constant.

Hence or otherwise, solve

Express 

in the form

where , and are integers to be found.

Hence sketch the curve with equation

showing the coordinates of the points at which the curve cuts the coordinate axes.

Use the factor theorem to show that  is a factor of the function

Hence express  as a product of three linear factors.

Sketch the curve  showing the coordinates of the points at which the curve meets the coordinate axes.

Sketch the curve with equation

showing the coordinates of the points at which the curve meets the coordinate axes.

Sketch the curve with equation

in the case where , stating the equations of any asymptotes.

You are given that  is inversely proportional to .

When , .

(i) Find an equation for in terms of .

(ii) Sketch the graph of against , where .

Sketch the curve with equation 

showing the coordinates of the points at which the curve meets the coordinate axes.

On the same diagram, sketch the curves with equations and .

Hence find the number of real solutions to the equation

Show that  is a factor of .

Hence express in the form

where , and are integers to be found.

Sketch the curve with equation

showing the coordinates of the points at which the curve meets the coordinate axes.

Sketch the curve with equation

stating the equations of any asymptotes.

On the same axes below, plot the curve  and the line .

Hence use a graphical method to solve the simultaneous equations

You are given that  is inversely proportional to .

When , .

(i) Find an equation for in terms of .

(ii) Sketch the graph of against , where .

The diagram below shows a sketch of the curve , where .

Sketch the graph of , where .

The graph passes through the point , where is a negative real constant.

Explain whether or .

Sketch the curve

showing the coordinates of any points at which the curve meets the coordinate axes.

On the same coordinate axes, sketch the curves and .

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

Figure 1 shows a sketch of a curve with equation where is a cubic expression in .

The curve

• passes through the origin

• has a maximum turning point at

• has a minimum turning point at

The line with equation , where is a constant, intersects at only one point.

Find the set of values of , giving your answer in set notation.

Find the equation of . You may leave your answer in factorised form.

Given that  is a factor of , sketch the curve

showing the coordinates of the points at which the curve meets the coordinate axes.

On the same diagram, sketch the curves with equations and .

Show the coordinates of any points where the curves meet the coordinate axes.

Hence find the number of solutions to the equation

By showing that  is a factor of the function 

fully factorise .

Sketch the curve , showing the coordinates of the points at which the curve meets the coordinate axes.

Find the coordinates of the points of intersection between the curve with equation 

and the line with equation

On the same coordinate axes, sketch  and .

Show the coordinates of

• any points where the curve meets the coordinate axes

• any points where the line meets the coordinate axes

• any points of intersection between the curve and the line

Figure 4 shows a sketch of part of the curve with equation

and part of the curve with equation

Verify that the curves intersect at

The curves intersect again at the point

Using algebra and showing all stages of working, find the exact coordinate of

You are given that  and are factors of , where

Sketch the curve  showing the coordinates of the points at which the curve meets the coordinate axes.

The curve has the equation

It is known that

• the -coordinate of the minimum point is positive

• the maximum point is in the second quadrant

By finding a factor of , sketch .

Show the coordinates of any points at which the curve meets the coordinate axes.

Fully factorise

Sketch the curve

showing the coordinates of the points at which the curve meets the coordinate axes.

On the same coordinate axes, sketch the curve 

and the line

Show the coordinates of

• any points where the curve meets the coordinate axes

• any points where the line meets the coordinate axes

• any points of intersection between the curve and the line