Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.

The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.

The upper bound for a rounded number is the smallest value that would round up to the next rounded value.

Signs of a solution

Maths

Checking and securing understanding of compound interest calculations

Building on composite functions

Evaluating iterative formulas

Approximating solutions to equations

Problem solving with iteration

In this unit, pupils continue to explore composite functions, thereby building on their work from the prior unit. Pupils need to understand function notation from the prior unit as this is used in iteration.

In this unit, pupils continue to explore composite functions and develop understanding of iteration, building on their work on functions and proof.

Iteration

There is a solution to the equation $$x^3-4x+2=0$$ between $$x=0$$ and $$x=0.5$$

There is a solution to the equation $$x^3-4x+2=0$$ between $$x=0.5$$ and $$x=1$$

There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1$$ and $$x=2$$

There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1$$ and $$x=1.2$$

There is a solution to the equation $$x^3-4x+2=0$$ between $$x=1.6$$ and $$x=2$$

Using the table of values shown for the equation $$y=x^3-4x+2$$ which of these statements are definitely true?

Between which sets of values is there definitely a solution to $$x^3-5x-1=0$$ ?

When the equation being solved is not equal to zero.

When one input is negative but the other is positive.

When the graph has an asymptote and doesn't cross the axis between those points.

When might a change in sign not indicate there is a solution between two values?

The image shows Jacob's workings to test his approximate solution to the equation $$x^3-2x=0$$. What do you think his solution was?

To see if $$x=-1.4$$ is a solution to the equation $$x^3-2x=0$$ to 2 significant figures which values should we test to see if there is a change in sign?

Which of these are solutions to $$-x^2+3x-1=0$$ correct to the stated degree of accuracy?

A value has been rounded to 2.8 to 2 significant figures. What is the lower bound of this value?

A value has been rounded to 2.8 to 2 significant figures. What is the upper bound of this value?

What is the value of $$x^3-2x+5$$ when $$x=2$$?

What is the value of $$x^3-2x+5$$ when $$x=-2$$?

Match the letter representing each graph to the equation of the graph.

Using the iterative formula $$x_{n+1}=\sqrt[3]{4(x_n)^2-1}$$ and $$x_0=5$$ there is a solution to $$x^3-4x^2+1=0$$ correct to 2 significant figures when $$x=$${{}} . 

Signs of a solution

Lesson outcome