Graphing Exponential Functions - Video Tutorials & Practice Problems

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Graphs of Exponential Functions

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Hey, everyone, whenever we graphed polynomial and rational functions, we had to consider a lot of different components. We looked at things like the end behavior and any intercepts of our graph and even had to consider and find multiple asymptotes sometimes when working with these functions. Now that we're faced with graphing exponential functions, you may be worried that there's going to be a lot of new and different things that we have to remember in order to fully graph them. But here we're going to walk through graphing the exponential function two to the power of X together. And I think you'll find that it's actually even more simple and more straightforward than graphing any polynomial or rational function. So let's go ahead and get started here. Now, here we're going to be graphing two to the power of X. So let's start by just plugging in some values for X and getting some ordered pairs to plot on our graph. So starting with X equals zero. If I plug that into my function, I get two to the power of zero, which is simply one because anything to the power of zero is one. So that first point I can plot is right at 01. Then moving on to X equals one plugging that into my function, I get two to the power of one, which is simply two. So that next point is at 12 that I can plot on my graph. Then as X continues to get larger, I'm going to get two squared which is simply four and then two cubed which is eight. And if I took it all the way up to X equals 10, I would get an even larger number this 1024. So I can see that as X continues to increase, F of X will continue to increase as well. So let's plot those last two points at 24 and at 38. Now looking at the side, I can go ahead and connect all of these points because I know it's going to increase. So I see that as X goes to infinity, F of X also goes to infinity. That's my end behavior on that side. Now let's consider if X is negative. So starting with X equals a negative one, if I plug that into my function, I get two to the power of a negative one, which knowing our rules for exponents, this is simply going to be 1/2. So that point is that negative 11 half. And then as X gets more negative negative two and negative three, it continues to get smaller. And smaller on that side. So we see that we're gonna get really close to zero but not quite zero. So I can go ahead and connect my points on this side. I see that I'm getting it really close to that X axis there. So as X approaches a negative infinity, I see that F of X actually approaches zero. Now, something I said there might have sounded familiar, we're going to get really close to zero but not quite touch it because that's what we know as an Asymptote. So we do have an Asymptote. It is a horizontal asym tote, right on that X axis at Y equals zero. Remember we can plot asymptotes by using a dashed line. So here I have my line right at Y equals zero. Now taking a full look at our graph here looking first at this right side, I see that we, as we increase, we look kind of similar to what we've seen with polynomial functions. Now, another similarity that our graphs of exponential functions have to polynomial functions is that they are always going to be continuous, meaning that there are no breaks in our graph. So we see that there are no breaks in this graph, it is continuous. Now looking at the other side of our graph here, as we approach that Asymptote, we know that we have asymptotes when dealing with rational functions. So another similarity that we may have to rational functions here is that our function is going to be 1 to 1 the way that some rational functions are. Now, this just means that no two X's result in the same Y or we pass the horizontal line test. If I draw a horizontal line, it's only going to cross through one point on my graph. Now, looking at the complete shape of the graph of my exponential function, this is is always going to be the shape and a way that you can remember this is exponential function starts with E. So if I take a lowercase E and I just extend that tail out, it looks really similar to the shape of this graph. Now that we have a complete picture of this graph. Let's also consider the domain and range here. So the domain of any exponential function is actually always going to be the same. It's always going to go from negative infinity to positive infinity or simply all real numbers. This is the same for any exponential function. Now your range is a little bit more complicated, but it just depends on your Asymptote. So it's always going to depend on the placement of our Asymptote here. And here my Asymptote is at zero and I see that all of my Y values are above that. So I know that my range here is going to go from zero to infinity but not including zero because I am of course not touching that Asymptote now that we have a good picture of our graph of two to the power of X. Let's take a look at some more exponential functions. Now, as we look at different exponential functions of the form F of X is equal to B to the power of X, we're going to see that the direction and the steepness of our graph is going to depend on the value of B the base of our function. So looking at if B is a greater than one, like something like two or five, I'm going to see that whenever I have that B greater than one, my graph is always going to be increasing and going up on that side. Whereas if B is in between zero and one, so something like one half or 1/4 I'm instead going to have a graph that decreases. Now, of course, also the steepness will depend on this value of B. And here we consider B greater than one. You see that as we increase from 2 to 5, our graph gets steeper. And that's because four values of B are based that are greater than one, like two or five, our graph is going to get steeper for larger values of B. Now, it's actually the opposite when working with value B in between zero and one, those fractions where we see that as we go from one half to 1/4 on our graph, as we get smaller, it actually gets steeper there. So for values of B in between zero and one. Our graph is actually going to get steeper for smaller values of B. So the opposite of if B is greater than one. So now that we have a complete picture of the graphs of our exponential functions. Let's get some more practice. Thanks for watching and let me know if you have any questions.

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example

Graphing Exponential Functions Example 1

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Hey, everyone in this problem, we're going to work through graphing the function F of X is equal to one half to the power of X together. So let's go ahead and get started. Now, we're just going to go through and plot some points by plugging in some values to our function. So let's start with X equals zero and plug zero into our function and see what we get. So if I take one half and raise it to the power of zero, I know that any thing raised to the power of zero is one regardless of if it's a fraction or not. So my very first point is at 01 and I can go ahead and plot that on my graph. Then if I plug one in for X, I get one half to the power of one, which if it's raised to the power of one, it just gets itself. So this is simply one half and I could plot my second point at 11 half. Then for two X equals two, plugging that into my function, I get one half a raise to the power of two. Now one raised to the power of two is just one. So really, I'm just doing 1/2 squared, which is going to give me 1/4. So we're getting much smaller here to 1/4 getting close to that X axis. And then finally, for X equals three, if I put one half and raise that to the power of three, that's going to give me 1/8, which is again getting really small and close to that X axis over here. Now looking at our negative numbers, if I start to plug in some negative numbers for X, starting with negative one, if I take one half and raise it to the power of negative one. Whenever we deal with whole numbers like two and raise them to a negative power, we end up with a fraction. Whenever we take a fraction and raise it to a negative power, we are instead going to end up with a whole number. So this is simply going to give me two because I just flip that fraction, I get 2/1, that's just two. So I can plot this point at negative 12, then if I take one half and raise it to the power of negative two, I'm going to get four. So I can plot this point at negative 24. Then for my last negative point, if I take one half and raise it to the power of negative three, I end up with eight and I can of course, go ahead and plot that point on my graph. Now, we see that we have all of these whole numbers as we get into the negatives. And then we had their corresponding fractions, we just flipped them under to the bottom of a fraction and they're the same numbers because we're dealing with the same powers, some are positive and some are negative. So some of these values will be fractions and some will be whole numbers just like we saw for two to the power of X. Now, if you think about it one half to the power of X is really just two to the power of negative X. So that's why we're seeing so many similarities here. Let's go ahead and connect our graph. So on this side going up and then on my right side here, I see that I'm getting really close to that X axis, not quite touching it though, which tells me I'm dealing with an Asymptote. So I can go ahead and plot my Asymptote here on this X axis because it is a horizontal Asymptote right at Y equals zero. And we can go ahead and to note that here, our Asymptote at Y equals zero, plotted, of course, using a dash line because it's an Asymptote. Now, looking at our graph here, you might notice that it looks really similar to the graph of two to the power of X, which looks something like this. So now our graph of one half to the power of X is kind of just this flipped, which makes perfect sense because we just said that this one half to the power of X is the same exact thing as two to the power of negative X. So it makes sense that it's the same graph but flipped because it just underwent a transformation due to this negative sign here. Now let's go ahead and finish up here and get the rest of the information for this graph like the domain and the range. So here our domain of every single exponential function is always going to be the same. So here it is the same as before all real numbers. Now, our range is dependent on where our Asymptote is. And since our Asymptote is again at Y equals zero, and our graph is completely above that we know that our range is going to go from our Asymptote at zero until infinity. So our range is zero to infinity using parentheses because that zero is not included. Now that we have the entire picture of our graph of one half to the power of X. Let's get some more practice.

3

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Transformations of Exponential Graphs

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Hey, everyone, we now know how to graph basic exponential functions of the form F of X is equal to B to the power of X where B is just some number like two or four or one half or whatever. But what if we're given a more complicated exponential function like this G of X that we have here, how are we going to graph that? We may be worried that we're going to have to take a bunch of different values for X and plug them in and do some messy calculations. But here I'm going to show you that we're not going to have to do a single cal because we can simply apply rules of transformations to our parent function B to the power of X that we already know how to graph. So all we're going to be doing here is taking the graph of a function that we already know how to get and picking it up and moving it around in order to get the graph of our new more complicated function. So let's go ahead and get started here. Now, when working with this G of X here, we see that it looks really similar to two to the power of X but just with a couple more things added in. So we have this negative one and this negative four, both of which represent transformations. Now, when working with exponential functions, we're going to be focused on two types of transformations in particular reflections and shifts because they are the most commonly occurring transformations when working with exponential functions. So just as a reminder whenever we have a negative on the outside of our function, that represents a reflection over the X axis, whereas a negative on the inside of our function represents a reflection over the Y axis. Now we're working with H and K, this H here represents a horizontal shift, remember H horizontal and then K of course represents a vertical shift by some number K units. So let's go ahead and start graphing our function at G of X here starting with our step zero. Now step zero is actually to identify and graph our parent function. And this step is really important because we always always want to graph our parent function first when transforming exponential functions. So let's go ahead and take a look at our G of X here. I said that it looks really similar to two to the power of X and that represents our parent function. So our parent function here is F of X is equal to two to the power of X. And let's go ahead and plot this using a couple of different points. So the first point that we have is going to be negative one, one over B. Now B is the base of our parent function, which here is two. So my first point will be at negative 1, 1/2, then my second point will be at 01, which will be the same no matter what your parent function is. And then finally, I'm going to plot one final point at one B which remember B is the base. So in this case too, so let's go ahead and plot these three points on our graph. So first negative 11 half, then 01 and finally 12. Now we wanna go ahead and connect all of these points using the shape of our exponential graph. And then we want to do one more thing to get that parent function completely graphed. And that is plot our horizontal Asymptote at Y equals zero. That Asymptote will always be at Y equals zero no matter what your parent function is. So we're using a dash line here because it is an Asymptote. Now that we have that parent function graphed, we can go ahead and move on to actually graphing our new transformed function G of X. So looking at that first step step one, we wanna go ahead and shift our horizontal Asymptote that we just drew. So looking at my function at G of X, I'm gonna wanna shift my horizontal Asymptote to Y equals K. Now, when looking at a transform function, I know that K is just added to the end of my function. And here I have this negative four. So I know that my K value is going to be negative four. So let's go ahead and shift that horizontal Asymptote all the way down to negative four. Of course, still using a dash line because it is an Asymptote. So with step one done, we have our horizontal Asymptote plotted. Let's move on to step two and determine if there is a reflection happening. Now, remember a reflection happens when we have a negative on the inside or the outside of our function and looking at my function at G of X, it looks like I did not have a negative get inserted into my new function. So I don't have to worry about a reflection and I can move on to the second part of step two part B to shift our test points by H and K. Now we're going to go ahead and identify H and K, we know that K is already negative four because we just moved our horizontal Asymptote. And then for H I have this X minus one here, remember when dealing with transformations, it's X minus H. So here my H value is simply one. Now we're going to go ahead and take those points from our parent function, our test points and shift them one over to the right and four units down in order to get the points of our new transformed function. So starting with this first point here, I'm going to go one over and then 1234 down to get my new point. Then for my 2nd 0.1 over again and then 1234 down to get my new point and then my last 0.1 over and then 1234 down to get my new point. So I have all of my new transformed points from my original function. And my very last step, my step three here is just to go ahead and connect everything. And of course, approach my Asymptote. So here I can go ahead and connect my points and then approach my Asymptote on this side. So here we can see that our original function just got picked up, it got shifted over and down a little bit in order to get our new function. So now that we have fully transformed our function, we have our graph of G of X, we can go ahead and identify our domain and our range. Now remember the domain of any exponential function is always going to be the same. It's always going to to go from a negative infinity to infinity or simply all real numbers. Now, for our range, this depends on our placement of our graph regarding our Asymptote. So here, if our graph is above our Asymptote, which here it is, we can go ahead and state that our range goes from K that value of K to infinity. Now here my value of K is negative four. So I know that all of my Y values are going to start at negative four and go to infinity. So here it is from K to infinity, negative four. Now, because this one is above, that's what our range is. But if it happened to be below our range, would instead go from negative infinity to K just depending on what's happening with our graph. Now that we know how to transform exponential functions. Let's get some more practice. Thanks so much for watching.

4

example

Transformations of Exponential Graphs Example 1

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4m

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In this problem, we want to graph our given function G of X as a transformation of F of X is equal to one half to the power of X. And the function I have G of X is negative one half to the power of X plus three. Now, the first thing that you might notice here is that we have this negative here. And I know that I've said before that the base of an exponential function cannot be negative and this actually does not violate that. So I have this negative outside these parentheses. If this was negative one half to the power of X, that's when we would have a problem. So just wanted to take a second to address that it's totally OK. If your negative is on the outside of your base, as long as it's not in there being raised to that power because here we're taking one half, raising it to the power of X and then making a negative. And that's what makes this OK. So let's go ahead and get right into graphing and start with step zero. Now, step zero is to identify and graph our parent function F of X is equal to B to the power of X. And we know here that our parent function is one half to the power of X. So going ahead and plotting the points that I need to, I want to graph negative 11 over B which is just negative 1201 and then one B which B here is one half. So let's go ahead and plot those three points. So negative 12 01 and then 11 half. And we can go ahead and connect all of these. And then of course, we want to plot our horizontal Asymptote at Y equals zero, which is just at our X axis using a dashed line. So we have our parent function here. Let's go ahead and get into our new function and transform this parent function. So our new function that we have is a negative one half to the power of X plus three. And the very first thing I wanna do is shift my horizontal Asymptote to Y equals K. Now, here I have this plus three on the end, which is my value for K. So I know that I need to have my horizontal Asymptote at Y equals three. So going ahead and putting our horizontal Asymptote using a dash line on my graph up here at Y equals three. Now that we're done with step one, we can move on to step two and decide if there is a reflection happening. Now, remember a reflection happens if we have a negative outside or inside of our function, and we do have this negative added to the outside of our function. So that tells us that yes, we do have a reflection which means that we need to take those test points from our parent function and reflect them here over the X axis because that negative is outside of our function. So let's go ahead and do that here taking my points. So starting with this point, I'm going to reflect that over the X axis. So my new point will end up here, same thing for this 01 here and end up right here and then my last point right here. So I have reflected my new points, but now I need to shift them. So remember we shift them by HK and here we don't have a value for H it's just zero. So I don't have to worry about any horizontal shift, but I do have a vertical shift by K. So we need to take all of these points and move them up by three. So let's go ahead and do that. So up by three shifting and then my other point here by three and then my last point up by three as well. So now I have my final points that I can go ahead and sketch and connect these points using a curve of course, approaching my Asymptote. So here we're approaching that Asymptote and then on this side as well. So we have our new function here, we have completed graphing it. But let's go ahead and identify this additional information. Our domain and range. Remember, our domain will always be the same no matter what, it's always going to be all a real number. So I don't have to decide anything or figure anything out there. But for my range, I need to look at whether my function is above or below the Asymptote. Now, since my Asymptote is right here, and my function is down here, that tells me that it is below my Asymptote. So my range is going to go from negative infinity until we reach that Asymptote at K, which in this case is at three. So here, my range is from negative infinity to three. Now that we've identified all of that information, let's get into some practice.

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Problem

Problem

Graph the given function.

$g\left(x\right)=4^{-x}-1$

A

B

C

D

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Problem

Problem

The graph for the function $f\left(x\right)=3^{x}$ is given below.

Match the given function, $g\left(x\right)$, to its graph. $g\left(x\right)=-3^{x+2}+1$