Cubic, reciprocal and exponential graphs

The expected value of a painting, £ P, is given by the formula

P = 2500 × 1.2ⁿ

where n is the number of years after it was bought and 0 ⩽ n ⩽ 4.

Write down the value of the painting when it was bought.

Write down the annual percentage increase in the expected value of the painting.

The table shows the expected value of the painting n years after it was bought.

On the page opposite, draw a suitable graph to show the expected value of the painting n years after it was bought, where 0 ⩽ n ⩽ 4.

Years after the painting was bought (n)

An art collector correctly works out 2500 × 1.2¹⁰ as 15479.

They say,

The expected value of the painting 10 years after it was bought is £15479.

What assumption has the art collector made.

Sketch the graph of y = 8^x. Indicate any values where the graph crosses the axes.

Sketch the graph of y = cos x for 0 ° ⩽ x ⩽ 360 °. Indicate any values where the graph crosses the x-axis.

The graph of y = 3^x is sketched below.

On the same axes, sketch the graph of y = 2^x.

Charlie sketches this graph.

Charlie says

The equation of my graph is y = sinx.

Explain how you know that Charlie is not correct.

Write down a possible equation for Charlie’s graph.

Sketch the graph of y = 5^x indicating any values where the graph crosses the axes.

Sketch the graph of y = tan x for 0 ° ⩽ x ⩽ 360 ° indicating any values where the graph crosses the axes.

The price, £ P, of a car is £20 000 in 2019. The price is expected to decrease by 5% each year after 2019.

Jasmine says

This means the price in 2021 is expected to be £18 000.

She is incorrect. Explain her error and work out the correct answer.

The correct answer is £........

Write a formula for P in terms of n, where n is the number of years after 2019.

Circle the graph that best represents the price, £ P, of the car n years after 2019.

Sketch the graph of y = 3^x. Give the value of the y-intercept.

Shirley is asked to sketch a graph of y = 5^x for x ⩾ 0. She produces the following.

The graph has two errors.

How should they be corrected?

For each graph below, select its possible equation from this list.

Amelia buys a new car. The expected future value of this car, £ V, is given by

where t is the age of the car in complete years.

Write down the value of the car when new.

Write down the annual percentage decrease in the expected value of the car.

Show that the expected value of the car when 2 years old is £9000.

Amelia sketches a graph to show the expected value of her car as it gets older.

Explain how you know that Amelia’s graph is incorrect.

Amelia assumes that her car will have no value at all after 20 years.

Explain why her assumption is mathematically incorrect.

Rashid invests money into an account which pays a fixed rate of compound interest each year. The value, £ V, of his investment after t years is given by the formula

How much money did Rashid invest?

What rate of compound interest is paid each year?

Circle the graph that best represents the growth in Rashid’s account.