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Exponential Graphs (AQA GCSE Further Maths)

Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Exponential Graphs

What is an exponential?

  • An exponential is a function where the power is a variable, usually x

    • y equals 3 to the power of x is an example of an exponential

  • In this course exponentials will be in either of the following forms

    • y equals a b to the power of x

    • y equals a b to the power of negative x end exponent

    • Where a and b are rational numbers, b space greater than space 0 and x is a variable

    • a can be equal to 1, resulting in y equals b to the power of x or y equals b to the power of negative x end exponent

    • All of the following are examples of exponentials you may encounter

      • y equals 2 cross times 0.3 to the power of x

      • y equals 5.2 cross times 3.8 to the power of negative x end exponent

      • y equals 4.1 to the power of x

      • y equals 0.34 to the power of negative x end exponent

What does an exponential graph look like?

  • A graph of the form y equals b to the power of x where b is positive and larger than 1 will be increasing as x increases

    • y equals 5 to the power of x is increasing

  • A graph of the form y equals b to the power of italic minus x end exponent where b is positive and larger than 1 will be decreasing as x increases

    • y equals 7 to the power of negative x end exponent is decreasing

  • If b is between 0 and 1, then the opposite is true

    • y equals 0.4 to the power of x is decreasing

    • y equals 0.6 to the power of negative x end exponent is increasing

  • An equation of the form y equals a b to the power of x stretches the graph of y equals b to the power of x vertically by scale factor a

    • If a is negative, then this would also reflect the graph in the x-axis

  • The bold italic y-intercept of y equals a b to the power of x and y equals a b to the power of negative x end exponent will be stretchy left parenthesis 0 comma space bold italic a stretchy right parenthesis

    • You can show this by substituting x equals 0 into the equation

    • Substituting x equals 0 into y equals a b to the power of x or y equals a b to the power of negative x end exponent will reduce both to y equals a cross times b to the power of 0 equals a cross times 1 equals a

    • This means that for an exponential in the form y equals b to the power of x or y equals b to the power of negative x end exponent, the y-intercept will simply be (0,1)

  • The graphs do not cross the x-axis anywhere

  • Exponential graphs do not have any minimum or maximum points

    • They are either always increasing, or always decreasing

Exponential Functions fig3, A Level & AS Maths: Pure revision notes

How can I find the equation of an exponential graph?

  • A typical exam question may give you one or two co-ordinates that lie on a curve, and an approximate form for the equation of the graph

    • e.g. y equals a b to the power of negative x end exponent or y equals k to the power of x

  • Remember that all co-ordinates on the curve must satisfy the equation

  • You can therefore substitute each coordinate into the given equation, and solve to find any unknown constants

Examiner Tips and Tricks

  • Remember that the y intercept can often be found by inspection, which may save you some working

    • For y equals b to the power of x or y equals b to the power of negative x end exponent the y-intercept is open parentheses 0 comma 1 close parentheses

    • For y equals a b to the power of x or y equals a b to the power of negative x end exponent the y-intercept is open parentheses 0 comma a close parentheses

Worked Example

Here is a sketch of the curve y equals a b to the power of negative x end exponent where a and b are positive constants.

open parentheses 0 comma space 6 close parentheses and open parentheses 2 comma space 0.375 close parentheses lie on the curve.

exponential-graphs-we-question

Work out the values of a and b.

 

The value of a can be found by inspection. The y-intercept is (0, 6) so a space equals space 6.

y space equals space 6 b to the power of negative x end exponent  

The value of b can be found by substituting the second coordinate into the equation and solving.

table row cell y space end cell equals cell space 6 b to the power of negative x end exponent end cell row cell 0.375 space end cell equals cell space 6 b to the power of negative 2 end exponent end cell end table

Solve to find b.

table row cell space 6 b to the power of negative 2 end exponent space end cell equals cell space 0.375 end cell row cell 1 over b squared space end cell equals cell space fraction numerator 0.375 over denominator 6 end fraction space equals space 1 over 16 end cell row cell b squared space end cell equals cell space 16 end cell row cell b space end cell equals cell space plus-or-minus 4 end cell end table

b must be positive, so disregard the negative value.

bold italic a bold space bold equals bold space bold 6 bold comma bold space bold italic b bold space bold equals bold space bold 4

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.