2.4 Further Functions & Graphs

Let

For the graph of f, find the equation of:

Find.

Write down the equation of the vertical asymptote to the graph of .

Let .

Find the coordinates of:

State the equation of the vertical asymptote to the graph of f.

The graph of intersects with its inverse, twice. 

Find the two coordinates where.

Let,  for .

On the following grid, sketch the graph of .

The inverse of f can be written in the form of .
Find the values of A, b and of c.

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, m g, in an object of age  years is

where and  are constants.

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

Briefly explain why, if ,  will equal g  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, V, of the vaccine in her blood over time can be modelled by an equation of the form , where V is the concentration (in mg) of the vaccine in the bloodstream and t 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, V₁ 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 V₂. took to reach a maximum volume compared to V₁.

Let , for . 

For the graph of f, find:

Find the value of

Given that , find the domain and range of g.

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 .

Solve the inequality .

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

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

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 following diagram shows part of the graph of  f  which crosses the x-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 a.

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

Find the value of p and the value of q.

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

Find the values of u, v and w.

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

where P₀ is the initial population and t is measured in years. 

After three years, it is estimated that .

Find the value of k and interpret its meaning.

Find the least number of whole years for which

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

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

Calculate the value of k.

State the domain and range of I.

Calculate the intensity of light 6.2 m below the surface.

Let , for , where .

The line x=2 is a vertical asymptote to the graph of

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

The line, where , intersects the graph of at exactly one point. Find possible values of .