What room Exponential Functions?

Before we obtain into handling exponential functions and graphing exponential functions, let's very first take a look in ~ the basic formula and also theory behind exponential functions.

Below is just one of the most general creates of an exponential graph:


A general instance of exponential graph

The exponential role equation come this graph is y=2xy=2^xy=2x, and is the most an easy exponential graph we can make. If you're wonder what y=1xy=1^xy=1x would look like, here's the exponential graph:


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Graph the y = 1^x

Now, as to the reason why the graphs that y=2xy=2^xy=2x and also y=1xy=1^xy=1x space so different, the best method to recognize the theory behind exponential attributes is to take a watch at part tables.

You are watching: The base of an exponential function can be a negative number.


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The table of worths of y = 1^x and y = 2^x

Above you deserve to see 3 tables because that three various "base values" – 1, 2 and also 3 – all of which room to the power of x. As you can see, for exponential features with a "base value" the 1, the value of y stays continuous at 1, since 1 come the strength of something is simply 1. That is why the above graph of y=1xy=1^xy=1x is simply a directly line. In the case of y=2xy=2^xy=2x and y=3xy=3^xy=3x (not pictured), top top the other hand, we check out an increasingly steepening curve because that our graph. The is due to the fact that as x increases, the value of y rises to a bigger and bigger value each time, or what we call "exponentially".

Now that we have an idea the what exponential equations look like in a graph, let's provide the general formula for exponential functions:

y=abd(x−c)+ky=ab^d(x-c)+ky=abd(x−c)+k

The above formula is a small more complicated than previous functions you've likely worked with, therefore let's define all of the variables.

y – the worth on the y-axis

a – the upright stretch or compression factor

b – the base value

x – the value on the x-axis

c – the horizontal translate in factor

d – the horizontal big or compression factor

k – the upright translation factor

In this lesson, we'll just be going over very basic exponential functions, so girlfriend don't need to worry around some the the above variables. But, so friend have accessibility to all of the info you need around exponential functions and also how to graph exponential functions, let's overview what an altering each of these variables does come the graph of an exponential equation.

1) variable "a"

Let's to compare the graph of y=2xy=2^xy=2x to an additional exponential equation wherein we modify "a", offering us y=(−4)2xy=(-4)2^xy=(−4)2x


A general instance of exponential graph
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compare the graph of y = 2^x and also y = (-4)2^x

By make this transformation, we have both "stretched" and "reflected" the initial graph that y=2xy=2^xy=2x by it's y-values. In stimulate to discover "a" by looking in ~ the graph, the most vital thing to notice is that as soon as x=0 and we don't have a worth for "k", the y-intercept of ours graph is constantly going come be equal to "a".

2)Variable "b"

Also recognized as the "base value" this is merely the number that has the exponent attached to it. Recognize it requires algebra, which will certainly be disputed later in this article.

Variable "c"

Let's compare the graph the y=2xy=2^xy=2x to an additional exponential equation whereby we change "c", providing us y=2(x−2)y=2^(x-2)y=2(x−2)


A general instance of exponential graph
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to compare the graph that y = 2^x and y = x^(x-2)

By do this transformation, we have actually shifted the whole graph to the right 2 units. If "c" was same to -2, us would have actually shifted the whole graph come the left 2 units.

Variable "d"

Let's to compare the graph that y=2xy=2^xy=2x to an additional exponential equation where we change "d", providing us y=24xy=2^4xy=24x


A general example of exponential graph
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to compare the graph of y = 2^x and y = 2^(4x)

By make this transformation, we have stretched the original graph the y=2xy=2^xy=2x by its x-values, comparable to exactly how the variable "a" modifies the function by that y-values. If "d" were negative in this example, the exponential role would experience a horizontal reflection together opposed to the vertical reflection seen through "a".

Variable "k"

Let's to compare the graph the y=2xy=2^xy=2x to another exponential equation where we change "k", providing us y=2x+2y=2^x+2y=2x+2


A general instance of exponential graph
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metric conversion table (length)

By make this transformation, we have actually translated the original graph the y=2xy=2^xy=2x up 2 units. If "k" were an unfavorable in this example, the exponential role would have been interpreted down two units. "k" is a specifically important variable, as it is likewise equal come what we speak to the horizontal asymptote! one asymptote is a worth for one of two people x or y that a duty approaches, however never in reality equals.

Take for instance the role y=2xy=2^xy=2x: for this exponential function, k=0, and also therefore the "horizontal asymptote" amounts to 0. This renders sense, since no issue what value we put in for x, we will certainly never obtain y to same 0. For our other role y=2x+2y=2^x+2y=2x+2, k=2, and also therefore the horizontal asymptote equates to 2. Over there is no worth for x we have the right to use to make y=2.

And that's all of the variables! Again, several of these space more complex than others, for this reason it will take time to obtain used come working through them all and also becoming comfortable recognize them. To get a better look at exponential functions, and to become familiar through the above general equation, visit this terrific graphing calculator website here. Take her time to play about with the variables, and get a far better feel for how an altering each that the variables impacts the nature that the function.

Now, let's get down come business. Offered an exponential duty graph, how deserve to we find the exponential equation?

How To find Exponential Functions

Finding the equation of exponential features is frequently a multi-step process, and also every trouble is various based upon the details and type of graph we space given. Provided the graph the exponential functions, we require to have the ability to take some info from the graph itself, and then deal with for the ingredient we are unable to take straight from the graph. Below is a perform of all of the variables us may need to look for, and how come usually uncover them:

a – deal with for it making use of algebra, or it will certainly be given

b – deal with for it making use of algebra, or it will be given

c – let x = 0 and imagine "c" is no there, the value of y will equal the y-intercept; now count how numerous units the y worth for the y-intercept is from the y-axis, and this will certainly equal "c"

d – solve for it using algebra

k – equal to the value of the horizontal asymptote

Of course, these are simply the basic steps you have to take in stimulate to discover the exponential duty equation. The best means to learn exactly how to perform this is to try some practice problems!

Exponential functions Examples:

Now let's try a pair examples in order to put all of the concept we've covered right into practice. Through practice, you'll be able to find exponential functions with ease!

Example 1:

Determine the exponential duty in the kind y=abxy=ab^xy=abx that the given graph.


finding an exponential function given that is graph

In stimulate to fix this problem, we're walking to need to find the variables "a" and "b". As well, we're going to need to solve both of these algebraically, together we can't recognize them native the exponential function graph itself.

Step 1: settle for "a"

To deal with for "a", we must pick a suggest on the graph wherein we can get rid of bx due to the fact that we don't yet understand "b", and therefore we need to pick the y-intercept (0,3). Because b0 equals 1, we can find that a=3. Together a shortcut, because we don't' have actually a value for k, a is simply equal come the y-intercept the this equation.


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discover a the the equation y = a b^x

Step 2: deal with for "b"

Now that we have "a", every we have to do is sub in 3 because that "a", pick one more point, and solve because that b. Let's choose the allude (1,6). V all this information, us can find that b=2.


Step 3: write the final Equation

Now we that we have found every one of the crucial variables, every that's left is to write out our final equation in the form y=abxy=ab^xy=abx. Our last answer is y=(3)2xy=(3)2^xy=(3)2x


Example 2:

Determine the exponential role in the kind y=a2dx+ky=a2^dx+ky=a2dx+k the the given graph.


In bespeak to fix this problem, we're walking to need to find the variables "a", "d" and also "k". Remember, us can uncover "k" native the graph, as it is the horizontal asymptote. For "a" and also "d", however, we're going to need to solve because that these algebraically, as we can't determine them from the exponential role graph itself.

Step 1: find "k" native the Graph

To discover "k", every we should do is discover the horizontal asymptote, which is plainly y=6. Therefore, k=6.


Step 2: solve for "a"

To deal with for "a", as with the critical example, we should pick a allude on the graph whereby we can remove 2dx due to the fact that we don't yet understand "d", and therefore we need to pick the y-intercept (0,3). Because 20 equates to 1, subbing (0, 3) into y=a2dx+6y=a2^dx+6y=a2dx+6 gives us the a=-3.

See more: How Many Vertices On A Triangular Prism Have? Triangular Prism


Step 3: solve for "b"

Now the we have actually "a" and "k", all we need to do is choose another suggest and resolve for b. Let's pick the allude (0.25, 0). V all this information, we can find that d=4.


Step 4: create the last Equation

Now us that we have found all of the important variables, all that's left is to create out our last equation in the kind y=abdx+ky=ab^dx+ky=abdx+k. Our final answer is y=(−3)24x+6y=(-3)2^4x+6y=(−3)24x+6


And that's it because that exponential functions! Again, these functions are a small more facility than equations because that lines or parabolas, for this reason be certain to do many practice difficulties to acquire a cave of the new variables and techniques. With much more practice, soon exponential equations and also the graphs that exponential functions will be no problem at all!