# Exponential Graphs

'Discover the properties of Exponential Graphs and pit your wits against the computer with some fun games'

Exponential functions are incredibly useful. The simplest 2, 4, 8, 16, 32, ..., doubling is probably the simplest case but they can be seen everywhere: describing growth in populations, money investments, decay of radioactive substances,... If you're attempting this activity you'll probably know a little about the shape of the exponential function, but this one takes it a little further. You will discover the effects of reflecting, stretching and translating have on these graphs. Play with the applets below and answer the quiz questions that follow then if you are up for the challenge, have a go at the two function games.

Watch this video to get an idea of what the task is about.

#### Resources

- Below there follows a set of three graphs to help you understand the properties of exponential functions and how reflecting, stretching and translating the graphs changes their equations.
- There is a quiz with each graph.
- Once you are confident you understand the properties you should try these games. Try to match the graphs by working out their equations.

#### Investigation - Reflections

Change the values of a and b by using the sliders to understand the properties of this function.

Check your understanding by completing this quiz.

**Exponential Reflections**

#### Investigation - Stretches

Change the values of a and b to understand the properties of the graph y = a^{bx}

Create the graph y = 2^{x}. How would you describe what happens to the gradient (slope) of the graph from left to right?

Make a graph that slopes **up** from left to right.

Change b to be negative. Make another graph that slopes **up** from left to right.

Make a graph that slopes **down** from left to right.

Change the sign of the value of b and make another graph that slopes **down** from left to right.

Make a horizontal graph.

Can you make a vertical graph?

Check your understanding by completing this quiz.

**Exponential Stretches**

- What is the equation of this graph?
- y=2^x
- y=2^2x
- y=-2^x
- y=2^-2x

- This is the graph of y=a
^{bx }Which of the statements

**best**describes what you can say about a and b?- 0<a<1
- b<0
- 0<a<1 AND b<0
- 0<a<1 OR b<0

- This is the graph of y = a
^{bx}Which of the statements best describes what you can say about a and b?

- a>1
- b>0
- a>1 AND b>0
- a>1 OR b>0

- It is possible to plot another graph that looks exactly the same as y = 2
^{x}. Which is it?- y=2^(-x)
- y=(-2)^(-x)
- y=0.5^(-x)
- y=-(2)^(-x)

*0.5*^{-1 }= 1/0.5^{1 }= 1/0.5 = 2

#### Investigation - Translations

Change the values of a and b to understand the properties of the graph y = 2^{x-a} +b. The asymptote of the graph and two points are plotted to help you understand the transformations.

Check your understanding by completing this quiz.

**Exponential Translations**

- The graph of y = 2
^{x-1 }+ 2 is plotted. What transformation would be viewed if I plot the graph of y = 2^{x-2 }+ 2 ?- Move 1 unit up
- Move 1 unit down
- Move 1 unit right
- Move 1 unit left

- The graph of Move y = 3
^{x + 2}- 1 is plotted. What transformation will be viewed if I plot the graph of y = 3^{x + 2}+ 1 ?- Move 2 units up
- Move 2 units down
- Move 2 units right
- Move 2 units left

- What is the equation of the following graph?
y = 3 ^{x + 2}+ 2y = 3 ^{x + 2}+ 3y = 3 ^{x + 2}- 3y = 3 ^{x - 2 }- 3A B C D - A
- B
- C
- D

#### Challenges

Once you are confident you understand the properties you should try these games. Your aim is to try to find the equation of the blue line. Type the equation in the input box and the red curve will match your equation. Try to make the red curve lie on top of the blue one. You should get a nice message if you get it right!

#### Challenge 1

Clue 1: Some points are plotted on the curve to help you.

Clue 2: The asymptote of the blue line is displayed.

Clue 3: The equation should be in the form y = a^{x – h + k}

Press refresh and you will get another equation. Repeat this at least five times until you are confident that you understand the properties of this type of graph.

#### Challenge 2

This one is a bit trickier and there are no clues!