2.5 Exponential Models

y = ka^x+c

The exponential function is set apart from other families of functions because of the fundamental impact of having the variable in the exponent. It stops being a multiplier and starts being more 'powerful' (forgive the bad joke). These functions can be counterintuitive as the speed with which their value increases can become extraordinary. One professor famously said that 'The biggest single failing of the human race is our inability to understand the exponential function. There is also a strong link here with Geometric sequences.

Key Concepts

In this unit you should learn to…

recognise exponential functions
find the y-intercepts of exponential functions
find the asymptotes of exponential functions

Essentials

Slides Gallery

Use these slides to review the material and key points covered in the videos.

What are exponential functions?

This is an introduction to exponential function and all of its unique properties.

Exponential y - intercepts

These are not as intuitive as they are with other functions because exponents behave differently when they are equal to zero! Watch to get a grip on this idea.

Understanding and finding asymptotes

Asymptotes are one of the more interesting features of exponential functions. Watch this video to understand them and how to find them.

Summary

This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.

Flash Card

Review this condensed 'key point' flashcard to help you check and keep ideas fresh in your mind.

Test yourself

Self Checking Quiz

Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.

Consider the exponetial function $f left parenthesis x right parenthesis space equals space 2 cross times 3 to the power of x plus 1$ ,

Fill in the corresponding values of the function for the given values of x.

x = 0, so f(x) =

x = 5, so f(x) =

Either substitue the value of x in to the function and calculate directly OR type the function in to your GDC and read your answers from the table. Note a⁰=1.

Consider the exponetial function $f left parenthesis x right parenthesis space equals space minus 1.5 cross times 2 to the power of x minus 2$ ,

Fill in the corresponding values of the function for the given values of x. (give exact answers)

x = 1 so f(x) =

x = -2, so f(x) =

Either substitue the value of x in to the function and calculate directly OR type the function in to your GDC and read your answers from the table. Note a⁰=1.

Write down the y-coordinate of the y-intercpt and the equation of the asymptote for the following exponential function.

$y equals 2 cross times 1.5 to the power of x plus 3.2$

The y-intercept is at (0, )

The asymptote has the equation y =

For the y - intercept, substitute x = 0. So, 1.5⁰= 1 therefore y = 2 + 3.2. SInce $2 cross times 1.5 to the power of x$ can never be zero, y can never be 3.2

Write down the y-coordinate of the y-intercpt and the equation of the asymptote for the following exponential function.

$y equals 1.2 cross times 3 to the power of negative x end exponent minus 18$

The y-intercept is at (0, )

The asymptote has the equation y =

For the y - intercept, substitute x = 0. So, 3⁰= 1 therefore y = 1.2 -18. SInce $1.2 cross times 3 to the power of negative x end exponent$ can never be zero, y can never be -18

Consider the graphs on the axes below. Write the corresponding letter next to each function. (use upper case)

y = 2^x,

y = 2^x+1,

y = -2^x,

y = 2^-x,

You can simply put these in your calculators. BUT you should recognises that A is the same as D but moved 1 up the y - axis. C has -x in the exponent to produce that reflection and B is -2^x , the negative of 2^x .

An experiment is done to monitor how a population of bacteria grows over time. At the start, there are 200 bacteria and they triple every hour. If the function we would use to model this is in the form $f left parenthesis x right parenthesis equals k cross times a to the power of x$ , what are the values of k and a?

k =

a =

200 must be the starting value, when x = 0. Next, 3 is the multiplier because the population triples for every x-increase by one.

The value of a car depreciates by 20% each year after purchase. It was purchased for $4000. If the function we would use to model this is in the form $f left parenthesis x right parenthesis equals k cross times a to the power of x$ , what are the values of k and a?

k =

a =

4000 must be the starting value, when x = 0. Next, 0.8 is the multiplier because the value becomes 80% of its previous value with each passing year.

Consider the model $T equals 59 cross times 1.8 to the power of negative x end exponent plus 15$ , which measures temperature (T) as time (x, in minutes) passes.

What is the initial temperature at the start of the experiment?

T =

This is when x = 0.

Consider the model $T equals 59 cross times 1.8 to the power of negative x end exponent plus 15$ , which measures temperature (T) as time (x, in minutes) passes.

What is the temperature after 4 minutes? (3sf)

T =

This is when x = 4. Read this from the table on your GDC.

Consider the model $T equals 59 cross times 1.8 to the power of negative x end exponent plus 15$ , which measures temperature (T) as time (x, in minutes) passes.

After how many minutes does the temperature fall below 18 degrees? (nearest minute)

After minutes.

Read this from the table on your GDC.

Exam Style Questions

The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.

Question 1

Video Solution

Question 2

Video Solution

Question 3

Video Solution

Question 4

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MY PROGRESS

Self-assessment

How much of 2.5 Exponential Models have you understood?

My notes

IBDP Maths: Applications & Interpretations SL

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2.5 Exponential Models

y = ka^x+c

Flash Card

Self Checking Quiz

IBDP Maths: Applications & Interpretations SL

Functions Home Free Access Weekend! Start here Examination Questions Practice Questions Question Bank Your Graphical Display Calculator Number and Algebra Geometry and Trigonometry Statistics and Probability Calculus Pre-IB Preparation Exams & Assessment For Teachers

2.5 Exponential Models

y = kax+c

Flash Card

Self Checking Quiz

Functions
Home
Free Access Weekend!
Start here
Examination Questions
Practice Questions
Question Bank
Your Graphical Display Calculator
Number and Algebra
Geometry and Trigonometry
Statistics and Probability
Calculus
Pre-IB Preparation
Exams & Assessment
For Teachers

y = ka^x+c