All you do to the graph to make

All exponential functions follow a basic graph. But when you make changes to the function, you will see the graph shift and make changes.

Watch this video lesson to learn how to easily identify these changes or transformations.

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The Basic Exponential Function

An exponential function is any function where the variable is the exponent of a constant. The basic exponential function is f(x) = b^x, where the b is your constant, also called base for these types of functions. Keep in mind that this base is always positive for exponential functions. All other exponential functions are based off of the basic exponential function.You will see that different exponential functions will add numbers to the basic exponential function in various locations, and these changes will produce changes in the graph as well.

When you graph any exponential function though, they will all have the same general look, based off of the basic exponential function graph. Depending on the change, or transformation, the graph can be shifted up or down, left or right, or even reversed. Keep watching, and we will identify what causes these changes.

First, let’s see what the basic exponential function looks like graphed. Before I show you how it looks, I want you to think about exponentials for a moment. What happens as you increase the exponent? Doesn’t the value of the function get increasingly bigger at an increasingly faster rate? It does, so you will see the graph curve upwards quickly.

And looking at our graph, we see that it does.

The basic exponential function.
exponential function

This particular graph shows the graph of f(x) = 2^x. Also notice that the graph crosses the y-axis when the exponent equals 0. For the basic function, it is when the x = 0 since our exponent is x by itself.

Now let’s see what happens to the graph when we change the function slightly.

Reversals

What can we change? Well, we can change the exponent to a negative so our function becomes f(x) = 2^(-x). What happens then? Think about what is happening to the exponent. The negative sign essentially reverses our variable. As you decrease the x-value and as it gets more negative, the function gets larger and larger more quickly. Why? Well, if we plug in a -2 for our x, the function becomes 2^-(-2) = 2^2 since the negative changes the sign of the exponent.

If we plug in a -3, the function becomes 2^-(-3) = 2^3. The more negative we get, the bigger our function becomes. So your graph flips or reverses itself.

Negative exponent reversal.

exponential function

Vertical Shifts

When you add or subtract a number from the basic function, we get vertical shifts. Think about this.

What can you do to the graph to make it go up or down? Wouldn’t you just have to add or subtract some numbers? If we added a 3 to our function to get f(x) = 2^x + 3, we would be shifting our graph 3 points upwards. You can think of this as adding 3 to every single point on our basic graph.

Shift up.
exponential function

If we subtract 4 from the function, what do you think will happen? Our graph will shift down by 4 points. We are essentially subtracting 4 from every single point of our basic graph.

Shift down.
exponential function

Also, both graphs cross the y-axis when x = 0 since the exponent is only x.

Horizontal Shifts

When we add or subtract from the exponent, the graph moves sideways. If we add a 2 to the exponent, we see the graph shifts 2 points to the left. Why? When we change the exponent, we are changing where the graph crosses the y-axis.

Remember that the graph crosses the y-axis when the exponent is 0. So, if our exponent has an added 2, we need to subtract 2 to get back to 0. Going left on the x-axis will give us that minus 2 that we need.

Shift left.

exponential function

If we subtracted 5 from the exponent, our graph would shift to the right by 5 points. Again, it is because the graph crosses when the exponent equals 0. When we subtract 5 from the exponent, we need to add 5 to get it back to where it normally equals 0, hence the shift of 5 to the right.

Shift right.
exponential function

Lesson Summary

Wow! Look at all we’ve learned! We’ve learned that an exponential function is any function where the variable is the exponent of a constant.

Our basic exponential function is f(x) = b^x, where b is our base, which is a positive constant. All other exponential functions are modifications to this basic form.Transformations are changes to the graph. Transformations include vertical shifts, horizontal shifts, and graph reversals. Changing the sign of the exponent will result in a graph reversal or flip. A positive exponent has the graph heading to infinity as x gets bigger.

A negative exponent has the graph heading to infinity as x gets smaller.Adding or subtracting numbers to the function will result in vertical, or up and down, shifts. Adding numbers shifts the graph up.

Subtracting numbers shifts the graph down. Adding or subtracting numbers to the exponent will result in horizontal, or sideways, shifts. Adding numbers to the exponent shifts the graph to the left, and subtracting numbers to the exponent shifts the graph to the right.

Learning Outcomes

Study and reference this lesson if you’d like to:

  • Recall the meaning of a basic exponential function
  • Interpret a graph shift along the x- or y-axis
  • Understand the transformation of a graph based on the modification to the original function
  • Note the correlation between a negative sign and the reversal of a variable
  • Distinguish between horizontal and vertical shifts
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