2.4 Other Functions & Graphs

Consider the functions and  .

Find the coordinates of the -intercepts for the graph of

Find the coordinates of the -intercepts for the graph of

For the graph of , find the equation of

the vertical asymptote

Consider the function  defined by , for , and the line 
The graph of  and the line intersect at points A and B.

Find the coordinates of A and B.

Find the midpoint of the line segment AB.

Let .

Find the coordinates of:

the intercept

the intercept

State the equation of the vertical asymptote to the graph of 

The graph of  intersects with its inverse, twice.

Find the two coordinates where .

Let .

On the following grid, sketch the graph of .

The inverse of  can be written in the form of .
Find the values of  and of 

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.
The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years.

A model for the mass of carbon-14,  g, in an object of age  years is

where  are constants.

For an object initially containing 100g of carbon-14, write down the value of 

Briefly explain why, if ,  will equal   when  years.

Using the values from part (b), show that the value of  is  to three significant figures.

A different object currently contains 60g of carbon-14.
In 2000 years’ time how much carbon-14 will remain in the object?

A small company makes a profit of £2500 in its first year of business and £3700 in the second year.  The company decides they will use the model

to predict future years’ profits.

 is the profit in the  year of business.

 and  are constants.

Write down two equations connecting and .

Find the values of  and .

Find the predicted profit for years 3 and 4.

Show that

can be written in the form

In an effort to prevent extinction scientists released some rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

 is the number of birds after  years of being released into the reserve.

Write down the number of birds the scientists released into the nature reserve.

According to this model, how many birds will be in the reserve after 3 years?

How long will it take for the population of birds within the reserve to reach 500?

Rebecca recently had the COVID-19 vaccine. The volume, of the vaccine in her blood over time can be modelled by an equation of the form  , where  is the concentration (in mg) of the vaccine in the bloodstream and  is time measured in days after 9am on Monday.

On the following grid, sketch the graph of 

Find, to the nearest minute, the time when the vaccine volume , reaches a maximum value.

Rebecca experienced side-effects from the vaccine between the times when the volume reached its maximum value until it had dropped to half of its maximum value. Find, to the nearest minute, the length of time that Rebecca experienced side-effects from taking the vaccine.

The vaccine is medically determined to be no longer in Rebecca’s bloodstream when it drops down to 1% of its maximum value. Find the time that the vaccine is no longer in Rebecca’s bloodstream.

Rebecca’s friend, Zara, also had the vaccine on the same day. The volume in Zara’s bloodstream can be modelled by an equation of the form of Calculate, to the nearest minute, how much faster  took to reach a maximum volume compared to .

Let  = and  = , where  and  is a constant.

Find .

Given that , find the value of .

Solve the equation 

Let , where and , a,b > 1.

The graph of  contains the points (0, 3) and (2, 75).

Find the values of  and .

Find an expression for 

Find the value of (375).

Consider 

Find the largest possible domain  for  to be a function.

Let , for  .

Explain why

is an even function

the inverse function  does not exist.

Let , for , and  for .

The graphs of  and  intersect at points  and .

Find the coordinates of  and .

Find the equation of the straight line at passes through  and , giving your answer in the form .

Write down the gradient of the line that is perpendicular to the line passing through  and .

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 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 .

olve the inequality .

Write down the domain and range of the logarithmic function  where and 

Given that , find all the expressions for  in terms of .

Let  where .

Solve the inequality .

For the graph of , find the coordinates of the

local maximum point.

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

Consider the function  defined by  for .

The following diagram shows part of the graph of  f  which crosses the -axis at point A, with coordinates (a, 0).  The line L is the tangent to the graph of  f  at the point B.

Find the exact value of .

The -coordinate of B is 10. The -coordinate of B can be written in the form , where .

Find the value of and the value of .

The gradient of L is  . The equation of L can be written in the form .

Find the values of and .

A population of endangered birds, , can be modelled by the equation

where  is the initial population and  is measured in years.

After three years, it is estimated that   .

Find the value of  and interpret its meaning.

Find the least number of whole years for which .

The intensity of light, , 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 a.

State the domain and range of .

Calculate the intensity of light 6.2 m below the surface.