Unit 2 - Functions
Get Ready
Look at the applet below.
Before revealing each graph, try to sketch what you think the graph will look like.
Think carefully about what changes each time and how this might affect the next graph.
These are all examples of exponential functions.
What can you say about exponential functions based on these graphs?
Notes
An exponential function is a function with the independent variable (usually \(x\) or \(t\)) in the exponent.
The most basic form is \(f\left(x\right)=a^{x}\), where \(a>0\) and \(a\ne 1\).
The magnitude of \(a\) affects the steepness of the cuve.
When \(a>1\), the graph shows growth and is increasing.
When \(0<a<1\), the growth shows decay and is decreasing.
The function \(y=a^{x}\) has a horizontal asymptote at \(y=0\), has a \(y\)-intercept at \(\left(0,1\right)\), and passes through the point \(\left(1,a\right)\).
A common adaptation of the exponential function is \(y=a^{-x}\). This flips when it is growth and decay.
Recall from the Laws of Exponents that \(2^{-1}=\frac{1}{2}\).
So \(a^{-x}=\left(\frac{1}{a}\right)^{x}\).
We often use the form \(y=a^{-x}\) when we have decay, using a value of \(a>0\) instead of a fractional value less than \(1\).
Investigation
Now consider the graphs in the applet below.
The general form of an exponential function is$$f\left(x\right)=k\times a^{x}+c$$where \(a>0,a\ne 1,k\ne 0\).
By investigating the graphs above, and others you create yourself, determine rules based on the values of \(a\), \(c\) and \(k\) for:
(a) the equation of the horizontal asymptote;
(b) the \(y\)-intercept;
(c) where the graph is in relation to the \(x\)-axis.
You can use the desmos graph below to further play around with exponential functions, or to check your answers to the following questions.
Examples and Your Turns
Example
Sketch the graph of \(y=5\times 2^{x}-3\), clearly indicating the equation of the asymptote and the coordinates of the \(y\)-intercept.
Your Turn
Sketch the graph of \(y=2\times 3^{x}+1\), clearly indicating the equation of the asymptote and the coordinates of the \(y\)-intercept.
Your Turn
Sketch the graph of \(y=3\times 2^{-x}+2\), clearly indicating the equation of the asymptote and the coordinates of the \(y\)-intercept.
Your Turn
Sketch the graph of \(y=-4\times 3^{x}-1\), clearly indicating the equation of the asymptote and the coordinates of the \(y\)-intercept.
Your Turn
Sketch the graph of \(y=3\times e^{x}-4\), clearly indicating the equation of the asymptote and the coordinates of the \(y\)-intercept.
Your Turn
Sketch the graph of \(y=-2\times e^{-x}+1\), clearly indicating the equation of the asymptote and the coordinates of the \(y\)-intercept.
Your Turn
For the graph below, state the \(y\)-intercept and the equation of the horizontal asymptote.
Use these to find the equation of the function in the form $$y=k\times 2^{\pm x}+c$$
Your Turn
For the graph below, state the \(y\)-intercept and the equation of the horizontal asymptote.
Use these to find the equation of the function in the form $$y=k\times 2^{\pm x}+c$$
Your Turn
For the graph below, state the \(y\)-intercept and the equation of the horizontal asymptote.
Use these to find the equation of the function in the form $$y=k\times 2^{\pm x}+c$$
Your Turn
For the graph below, state the \(y\)-intercept and the equation of the horizontal asymptote.
Use these to find the equation of the function in the form $$y=k\times 2^{\pm x}+c$$
Your Turn
Find the equation of the function shown in the graph below.
Your Turn
A scientist monitoring a grasshopper plague notices that the area affected by the grasshoppers is given by \(A\left(n\right)=1000\times \left(1.15\right)^{n}\) hectares, where \(n\) is the number of weeks after the initial observation.
a) Find the original affected area.
b) Find the area affected after (i) 5 weeks (ii) 10 weeks.
c) Sketch the graph of the affected area over time.
d) Use your graph to find how long it will take for the affect area to reach \(8000\) hectares.
e) Comment on the validity of this model.
Your Turn
The temperature of a cup of coffee is modelled by the equation \(T\left(x\right)=65\left(1.9\right)^{-x}+20\), where \(T\) is the temperature in \(^{\circ} C\), and \(x\) is the time in minutes after the cup of coffee was poured.
a) Find the original temperature of the coffee.
b) Determine the temperature of the coffee after \(3\) minutes.
c) Sketch the graph of Temperature over time.
d) State the equation of the asymptote.
e) Explain the significance of the asymptote in this scenario.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why do we define that \(a\ne 1\) for an exponential function \(y=a^{x}\)?
Why must \(a\) be positive? What would happen if \(a\) were negative?
Why does the graph only approach the asymptote on one side?
Common Mistakes / Misconceptions
Believing that the graph eventually touches the asymptote. It never reaches the asymptote, but gets closer and closer.
Connecting This to Other Skills
This skill builds upon the ideas of exponentials (indices) in the prior knowledge topics, as well as logarithms (1.1). You might need to solve equations with exponentials (1.12) for some problems.
When we get to Function Transformations (2.18) we will see how the different forms all stem from the original basic function that is \(y=a^{x}\).
In Differentiating Exponentials and Logarithms (4.12) we will see more properties of these graphs and apply the principles of calculus to exponential functions. We continue with this idea in the Reverse Chain Rule (5.4) and when we get to Separable Differential Equations (8.5)
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?