Further Functions & Graphs (DP IB Maths: AI HL)

Let . The diagram below shows part of the graph of .

Another function is defined by 

Sketch the graph of  on the axes above.

The graph of and  intersect at points A and B.

Find the coordinates of A and B and label them on the diagram above.

Find the length of the line AB.

Let , for .

For the graph of , find the:

-intercept

-intercept.

For the graph of , write down the equation of any asymptotes.

Let for  The graphs of  and  intersect at points  and .

Write down the coordinates of P and Q.

Find the distance of .

The perimeter, P, and area, A, of a given square can be expressed by and  respectively, where  is the length of the side of the square.

Write down an expression for:

P in terms of A,

A in terms of P,

Find the value of  and .

Consider the function  where  and  are constants. The graph of  passes through the points and     and is shown below.

Write down two equations relating and .

Find the value of and .

Write down the equation of the horizontal asymptote of the graph of .

The function  intercepts the -axis at  and has an -intercept at  The function can be obtained by an appropriate shift of the graph .

Find the values of  and .

Find the other -intercept of .

Determine the coordinates of the maximum value of .

A function is defined by 

Find the value of 

For the graph of  write down the equation of any asymptotes.

Find the range of .

The line  intersects the graph of  when  and when

Find the equation of . Give your answer in the form , where and  are integers.

A function is defined by 

Find the value of . Give your answer as a fraction.

Find the range of .

Find the value of . Give your answer as a fraction.

The diagram below shows the graph of the function , for .

For the graph of , find the:

-intercepts

-intercept.

Write down the range of .

Three lines are drawn connecting the two local minimum points and the local maximum point forming a triangle. Calculate the area of the triangle.

Let   for 

The th maximum point on the graph of  has  coordinate , where .

Given that , find and .

Using sigma notation, write down an expression for .

Find the value of the sum from part (b) (i). Give your answer to 2 decimal places.

The average fat-free mass, M, in kg, of footballers as a function of their age, , in years, can be given by the logarithmic function:

Calculate the average fat free mass of players aged:

Find an expression for a linear model using your answers to part (a) (i) and (ii).

Calculate the percentage error from using the linear model found in part (b) to approximate the average fat free mass of a player aged 20 years old.

The axes below shows the graph of the piecewise function, 

The gradient of the graph of  is 0 for .

Find the values of  and .

Complete the following piecewise function for 

A function is defined by . The graph of  has an axis of symmetry of 

Find the value of .

Find the range of .

Another function is defined by . The graph of  and  intersect at points A and B.

Find the equation of the line passing through points A and B. Give your answer in the form 

Find the distance of the line AB.

Consider the function  The line  intersects the graph of at point A and B.

Find the value of  and .

Find the equation of . Give your answer in the form , where  and  are fractions.

The line  is the perpendicular bisector to [AB].

Find the equation of . Give your answer in the form , where and are integers.

A rectangle has length  and width .

Find an expression for

the perimeter of the rectangle, P, in terms of .

the area of the rectangle, A, in terms of .

Show that .

The graph of the function P, for , is shown below.

On the grid above, draw the graph of the inverse function .

The function is a quadratic in the form , for

The graph of  has -intercepts  and 

Find the values of  and 

Another function can be defined by , for .

The graph of  and  intersect at points  and .

Find the coordinates of and .

Solve the inequality .

Let  for 

The th minimum point on the graph of  has  coordinate , where .

Given that , find .

Let  where .

Solve the inequality .

For the graph of , find the coordinates of the

Write down the possible domains of  for which  has an inverse and explain why the domain must be restricted.

The graph of a quadratic function has equation , where  and the axis of symmetry is .

Draw the axis of symmetry on the grid above.

The graph of the quadratic function intersects the -axis at points  and B.

Write down the coordinates of B.

Find the values of  and .

The diagram below shows part of the graph of the function , where points A, B, C and D represent stationary points. The points A, B, C and D are joined together forming a quadrilateral ABDC.

Write down the coordinates for

.

.

Find the area of the quadrilateral ABDC.

The intensity of light, I, is assumed to be 100% at the surface of the ocean and decreases with depth, , and can be estimated by the function

where  is expressed as a percentage,  is the depth below the surface, in metres, and  is a constant.

Calculate the value of .

State the domain and range of .

On a particularly bright day the intensity of light 6.2 m below the surface is 75%.

Calculate the percentage error between the estimate made by the function and the actual intensity of light.

Graph the following piecewise function on the grid below

Find the values of  when