Exponential: y=a^x (Junior Cert Mathematics): Revision Notes

Exponential Functions

An exponential function is a special type of function in maths where the variable (usually $x$) appears as the exponent (or power). Don't worry if that sounds a bit tricky—let's break it down!

For example, look at the function:

$f(x) = 2^x$

Here:

• The number $2$ is called the base.
• The letter $x$ is the exponent or power. So, in this function, the base $2$ stays the same, but the exponent (which is $x$) can change. That's what makes exponential functions unique!

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What Happens When You Change $x$?

When you change the value of $x$, the output (or result) of the function changes too. Here are some examples:

• If $x = 0$, then $f(x) = 2^0 = 1$ (Remember, any number to the power of $0$ is $1$!)
• If $x = 1$, then $f(x) = 2^1 = 2$
• If $x = 2$, then $f(x) = 2^2 = 4$
• If $x = 3$, then $f(x) = 2^3 = 8$ Notice how quickly the numbers get bigger as $x$ increases? That's what makes exponential functions grow so fast.

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How to Draw the Graph of $f(x) = 2^x$

To draw the graph of this function, let's make a table with some values of $x$ and the corresponding values of $f(x)$:

$( x )$	$f(x) = 2^x$
-2	0.25
-1	0.5
0	1
1	2
2	4
3	8

Now, plot these points on graph paper or using a grid:

• When $x = -2$, $f(x) = 0.25$ (very small)
• When $x = 0$, $f(x) = 1$
• When $x = 3$, $f(x) = 8$ (much bigger) Join the points with a smooth curve. You'll see that the graph starts out small on the left, rises through the point $(0,1)$, and then curves upward more steeply as you move to the right.

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Key Features of Exponential Graphs:

1. Starts Slowly, Then Grows Quickly: The graph starts out increasing slowly for smaller values of $x$, but as $x$ increases, the graph shoots up quickly.
2. Always Positive: The graph never dips below the $x$-$axis$, which means the values of $f(x)$ are always positive.
3. Passes Through $(0,1)$: All exponential functions of this type will cross the $y$-$axis$ at the point $(0,1)$.