Algebra 2 Exponential Functions

Welcome to Educator.com.0000

Today begins the first in a series of lectures on exponential and logarithmic relations, starting out with exponential functions,0003

beginning with the definition: What is an exponential function?0012

Well, an exponential function is a function of the form f(x) = a times b^x, where a is not 0,0016

because if a were to equal 0, this would all just drop out, and you wouldn't have a function.0027

b is greater than 0: we are restricting this definition to values of b that are positive; and b does not equal 1.0032

If b were to equal 1, no matter what you made x, that would still remain 1.0039

And then, you wouldn't really have a very interesting function.0044

So, the base of the function here is b.0047

Now, recall earlier on, when we worked with functions and equations, we have seen things like this: x² + 2x - 1...f(x) equals this.0050

Here, the base was a variable, and here the exponent was a constant.0062

Now, we are going the other way around: in this case, with exponential functions, we are going to be working with situations0070

where the base is a number (a constant) and the exponent contains a variable.0084

And that is what makes these functions fundamentally different from some of the other functions that we have seen so far.0101

Examples would be something like f(x) = 6x: here the base is 6, the coefficient is just 1 (a = 1), and the exponent is x.0110

Or you could have something a little bit more complicated: f(x) = (3 times 1/2)^{4x - 2}.0128

I have an algebraic expression, not just a single variable, as the coefficient.0138

So here, I have a base equal to 1/2; the coefficient is 3; and then the exponent is 4x - 2.0142

Starting out by looking at the graphs of exponential functions: as we have done with other types of functions,0153

we can use a table of values to graph an exponential function.0160

And we are going to look at a few different permutations of these functions and see what we end up with.0163

I am going to start out with letting f(x) equal 3^x; let's find some values for x and y.0170

I am going to just rewrite f(x) as y.0180

If x is -3, then what is y? Well, this is giving me 3 to the -3 power.0190

Recall that a^-n equals 1/aⁿ; so what this is really saying is 1/3³, which equals 1/27.0197

At this point, if you are not really comfortable with exponents and the rules and properties governing working with exponents,0215

you should go back and review the earlier lecture on that, because we are going on to solve equations using these.0220

So, you need to have the rules learned, as we will be applying them frequently.0227

When x is -2, this is going to give me 3^-2, which is 1/3², or 1/9.0231

And we continue on: when x is -1, that is 3^-1 = 1/3¹, or 1/3.0239

Then, getting back to some more familiar territory: when x is 0, this gives me 3⁰.0248

A number or variable, or anything, to the 0 power, is going to give me 1; when x is 1, f(x), or y, is 3.0257

When x is 2, that is 3² is 9; when x is 3, that is 3³, to give me 27.0266

Let's go ahead and plot these values out; let's do that so that this is -2, -4, -6, -8, so we can look at more values right on this graph.0276

2, 4, 6, 8, 2, 4, 6, 8, 10; -2, 4, 6, 8, 10; so -10 is down here.0288

When x is -3, the graph is just slightly above 1; it is 1/27; when x is -2, we get a little farther away from 1--it becomes 1/9.0299

When x is -1, the graph rises up even a little more and becomes -1/3.0312

When x is 0, y is 1; when x is 1, y is 3; when x is 2, y is right up here at 9; and then, when x is 3, y gets very large.0321

This gives me a sense of the shape of the graph--that as x becomes positive, y rapidly becomes a very large number;0336

as x is negative, what I can see happening is that f(x) is approaching 0, but it never quite gets there.0345

Therefore, what I have is that, for this graph, the x-axis is a horizontal asymptote.0358

So, this is the graph of f(x); let's look at a different case--let's look at the graph of...this is in the form f(x) = ab^x;0377

let's look at the graph f(x) = a(1/b)^x.0392

Well, a is just 1, so I am going to look at the situation g(x) = 1 over 3 to the x power.0398

I am making a table of values, again, for x and y.0410

Again, when x is -3, let's look at what y is; it is going to be 1/3 to the -3.0419

Using this property, I am then going to get 3³, which is 27.0426

For -2, I am going to get (1/3)^-2, or 3², which is 9.0433

-1 is going to give me (1/3)^-1 = 3; 0...anything to the 0 power is simply going to be 1.0440

(1/3)¹ is 1/3; (1/3)² is 1/9; and (1/3)³ is 127.0453

Let's see what happens with this graph: here, when x is -3, y is very large--it is some value way up there, 27 (that is off my graph).0469

Let's look at -2: when x is -2, y is 9 (that is right here).0479

And I know that, when I get out to -3, this is increasing; so I know the shape up here.0484

When x is -1, y is 3; when x is 0, y is 1; they have the same y-intercept, right here at (0,1).0489

When x is 1, y is 1/3; when x is 2, we see that the graph is approaching the x-axis, because now, we are here at 1/9.0502

And when x is 3, the value of the function is 127.0515

So again, I see this x-axis, again, being the horizontal asymptote for both graphs.0519

I know that y is getting very large as x is becoming negative.0532

And I know that, as x is positive, the graph is approaching, but not reaching, the x-axis.0538

Here, the y-axis also forms an axis of symmetry; so these two graphs are mirror images of each other.0547

f(x) and g(x) are mirror images reflected across the y-axis.0553

Let's look at one other case: let's look at the case...let's call it h(x) = 3^x, but let's take the opposite of that.0558

Therefore, this will be simpler to figure out the values: let's just leave the x-values the same as they were for f(x),0569

and let's just extend this graph out: these were my values for f(x), and now let's figure out what h(x) is going to give me.0581

Well, if x is -3, and I put a -3 in here, again, I am going to get 1/27, but I want the opposite of that; so I am going to get -1/27.0592

If x is -2, again, I am going to get 3^-2, which is 1/9, but I am going to take the opposite.0602

So, all I have to do is change the signs on these to get my h(x) values.0608

And then, we can see what this graph looks like.0615

This is f(x); I have g(x) here; for h(x), when x is -3, h(x) is right here; it is very close to the x-axis, but not quite reaching it.0620

And at -2, it is a little bit farther away at -1/9; at -1, we are down here at -1/3; at 0, here we have the y-intercept at (0,-1).0633

When x is 1, y is -3, right about here; when x is 2, y is down here at -9; and when x is 3, we are going to be way down here at -27.0649

So, for h(x), what I am going to have (I'll clean this up just a bit)...0664

This is the graph of h(x), and again, I am seeing the x-axis here, acting as a horizontal asymptote.0686

And I am seeing that, as x becomes positive, y becomes very large, but this time it is in the negative direction.0699

It is giving me a mirror image here with f(x); but now, my values are going down in the negative direction.0706

So, these are several graphs of exponential functions; let's go ahead and sum up the properties of these functions.0712

What we saw is that f(x) is continuous and one-to-one.0721

I am just very briefly sketching these three situations: I let f(x) equal 3x, g(x) equal (1/3)^x, and h(x) equal (-3)^x.0724

So, for the graph of f(x), what I got is this...actually, rising faster; so let's make that rise much faster.0746

And I saw that it is continuous; there weren't any gaps or discontinuities.0759

When we worked with graphing some rational functions, we saw that there were actually discontinuities; there aren't any here.0765

It is also one-to-one; and I could use the vertical line test.0771

Recall that, if you draw a vertical line across the curve of a graph, if it only crosses the graph once,0774

at one point, everywhere you possibly could try, you have a one-to-one relationship; you have a function.0782

So, no matter where I drew a vertical line, I would only cross this curve one time.0790

The domain of f(x) is all real numbers: you see that I could make x a negative value; I could make it a positive value.0795

However, the range is either all positive real numbers or all negative real numbers.0802

Here, the range is positive real numbers; for h(x), I graphed that, and that turned out like this.0807

Here, the domain was all real numbers, but the range was the negative real numbers.0820

We saw that the x-axis acts as an asymptote that the graphs of these functions approach, but never reach.0835

The y-intercept is at (0,a); so for both g(x) and f(x), here a = 1, so the coefficient is 1; so the y-intercept is at (0,1).0842

However, for h(x), a = -1; the coefficient is -1, so I have a y-intercept here at (0,-1).0861

Finally, we saw that the graphs of f(x) = ab^x and f(x) = a/(1/b^x) are reflections across the y-axis.0872

So, that was this graph, which shares the same y-intercept, but is reflected here across the y-axis.0880

So, looking at these three different situations and their graphs can tell us a lot about what is going on with exponential functions.0893

Introducing the concept of growth and decay: if you have a function in this form, f(x) = ab^x,0907

and a is greater than 0 (it is a positive number), and b is greater than 1, then this represents exponential growth.0916

And the concepts of exponential growth and decay are very frequently used in real-world applications.0925

So, we are going to delve into this topic in greater depth in a separate lecture.0931

But thinking about exponential growth: that could be something like working in finance, or thinking about your own savings and compound interest.0937

That works in such a way that the growth is exponential.0947

An example of exponential growth would be something like f(x) = 1/4(2^x).0952

And that is because here, b is greater than 1; so this is growth.0966

If I had another function, f(x) = 4((1/8)^x), here this is representing exponential decay,0972

because b is greater than 0, but it is less than 1: it is a fraction between 0 and 1.0983

And we sometimes talk about things such as radioactive decay and half-life in terms of an exponential function.0990

Another way to think of this for a minute is: recall that a^-n = 1/aⁿ.0999

So, I can actually rewrite this function as 4(8^-x).1006

If I had a base here that was greater than 1, but the exponent was negative, then I also know that I have decay.1014

If you write it in the standard form where the exponent is positive, then all you need to do is look and see the value of b.1022

If it is greater than 1, you have growth; if it is between 0 and 1, you have decay.1029

If it is a number greater than 1, and it is not in standard form, and you see I have a negative exponent,1033

that also gives you a clue that you are looking at decay.1039

But you can always put them in standard form, like this, and then just go ahead and look at the value of the base.1043

Working with exponential equations: in exponential equations, variables occur as exponents.1053

That is what I mentioned in the beginning, when I was talking about exponential expressions and exponential functions.1059

But now, we are talking about equations: again, you are used to working with things where the base may be a variable, but the exponent is a number.1064

And now, we are going to change that around and actually have situations where variables are exponents.1072

There are some properties of this that we can use to solve these equations.1079

If the bases are the same (if b^x equals b^y), then x must equal y.1083

If the bases are the same, in order for the left half of the equation to equal the right half, the exponents have to be the same; they have to be equivalent.1090

Look at this in a very simple case: 6^{3x - 5} = 6⁷.1100

6 is the same base as 6; so for this left half to equal the right half, if the bases are equal, these two must be equal.1108

So, in order to solve these, I am just going to take 3x - 5 and put that equal to 7, which leaves me with a simple linear equation to solve.1116

I am going to add 5 to both sides, which is going to give me 3x = 12.1125

Then, I am going to divide both sides by 3 to get x = 4.1129

Things get a little more complicated if the bases are not the same.1137

If the bases are not the same, the first thing to do is try to make them the same.1141

If that is not done reasonably simply, then we can use another technique that we are going to learn about in a later lecture.1145

But for right now, we are going to stick to situations where either the bases are the same, or you can pretty easily make them the same.1152

For example, I could be given an exponential equation 2^{x - 3} = 4.1159

These bases are not the same, so I can't use this technique.1168

However, you can pretty easily see that you can make them the same base1171

by saying, "OK, 2 squared is 4; so instead of writing this as 4, I am going to write it this way."1175

Now, I am back to the situation where the bases are the same; I am going to set the exponents1185

equal to each other (because they must be equal to each other) and solve for x: x = 5.1191

You could have a little bit more complicated situation, where 3^{x - 1} (a separate example) = 1/9.1199

And I can see that I want these to be the same, and I know that 1/3 squared is 1/9.1208

So, I am pretty close; but I need this to be 3.1214

Well, recall that I could rewrite this as 3^-2: 3^-2 is the same as 1/3 squared.1217

This also equals 1/9; and it is fine that this is a negative exponent--I can go ahead and use it up here, rewriting 1/9 as 3^-2.1227

Now, -x - 1 = -2, so x = -1.1236

OK, so basically, when you are working with exponential equations,1252

if they are the same base, you simply set the exponents equal to each other,1259

because this property tells us that that must be the case.1268

If the bases are not the same, try to make them the same: that is going to be your first approach.1272

And then, once you have written them as the same base, then you go ahead and solve by setting the exponents equal to each other.1278

Now, let's look at exponential inequalities: exponential inequalities involve exponential functions.1289

We just talked about exponential equations, and this is a similar situation, except we are working with an inequality, not an equation1296

(greater than, less than, greater than or equal to, less than or equal to).1303

And there are some properties that we can use to help us solve these.1307

If b > 1, then b^x > b^y if and only if x > y.1311

And this makes sense: if I have the same base (these are the same),1320

the only way that this left half is going to be greater than the right half1325

is if these exponents hold the same relationship, where x is greater than y.1330

And this could be greater than or equal to, or less than, or less than or equal to.1335

So, b^x < b^y if and only if x < y--the same idea.1339

If the bases are the same, and this on the left is less than the one on the right,1346

then that relationship, x < y, must be holding up.1350

We are going to use this property to solve inequalities.1355

For example, 4^{x + 3} > 4²: I know that the bases are equal, so that x + 3 must be greater than 2.1357

So, I just solve, subtracting 3 from both sides to get x > -1.1371

Again, if you are trying to work with a situation where you are solving an exponential equation or inequality,1378

and the bases aren't the same, see if you can make them the same.1384

5^6x < 115 well, I know that 5² is 25; if I multiply 25 times 5, I am going to get 125.1389

Therefore, 5² times 5...that is 5³...equals 125.1404

I am going to rewrite this as 5³; now that the bases are the same, I can say, "OK, 6x is less than 3, so x is less than 1/2."1410

Again, the idea is to get these in the form where the base is the same,1426

and then use the property that the relationship between the exponents has to be maintained,1431

according to the inequality, if the bases are the same.1439

Looking at examples: let's go back to talking about graphing.1446

We are asked to graph this function, f(x) = 3(2^x)--graphing an exponential function.1449

x...and we need to find y, so that we can do some graphing.1457

When x is -3, this is going to be 3 times 2 to the -3 power.1467

Again, this is equal to 3 times 1 over 2³; this is going to be equal to 3 times...2 times 2 is 4, times 2 is 8; or 3/8.1474

When x is -2, I get 3 times 2^-2; 3 times 1/2 squared...2 times 2 is 4, so this gives me 3 times 1/4, or 3/4.1492

-1: 3 times 2^-1 equals 3 times 1/2¹...remember that this -1 tells us that we need to take 1/the first power.1509

So, that is 3 times 1/2, which is 3/2.1530

0 gives me 3 times 2⁰, or 3 times 1, which equals 3.1537

Using 1 as the x-value gives me 3 times 2¹, or 3 times 2, which equals 6.1546

Using 2 gives me 3 times 2 squared, which is 3 times 4, or 12.1555

And then, one more: 3 times 2 cubed equals 3 times 8, or 24.1562

Plotting these values: when x is -3, y is 3/8 (just a little bit greater than 0); when x is -2, y gets a little bit bigger; it is 3/4.1570

When x is -1, then y becomes 3/2, or 1 and 1/2; 3/2 is going to be about here.1596

When x is 0, y is 3; when x is 1, y is up here at 6, rising rapidly; when x is 2, y will be all the way up here at 12.1607

I have enough points to form my curve; and I see the typical properties of exponential functions.1620

Remember that the y-intercept is going to be at (0,a): in this case, a = 3, so my y-intercept is going to be at (0,3), and that is exactly what I see.1629

And I can see that over here in the table of values.1646

I also see that the x-axis is an asymptote, and that the graph is approaching, but never reaching, the x-axis.1648

And I also can see that, as x becomes large in the positive direction, the value of y rapidly increases.1659

This is a graph of a typical exponential function.1667

Example 2: Does this function represent exponential growth or decay?1671

Recall that, if a is greater than 0, then we can look at a function in the form f(x) = ab^x and evaluate b.1677

If the base is greater than 1, then we have exponential growth.1691

If b is greater than 0, but less than 1, we have exponential decay.1698

But the caveat is that it has to be in the standard form; and I have an equation here that is not in standard form.1704

But I can use my rule that a^-n equals 1/aⁿ.1711

Therefore, I am going to rewrite this as f(x) = 4((1/5)^x).1716

So, I just took the reciprocal of the base and rewrote it.1727

And this is talking about a different 'a' than I am talking about here, with the coefficient: here, the coefficient is a.1735

All right, now that I have it in this form, I can evaluate; and I see that my coefficient is greater than 0, and that b is actually between 0 and 1.1742

Therefore, this is exponential decay.1751

I also could have looked back at the original and recalled that, if b is greater than 1,1756

and then you have a negative exponent, that would also give me a clue that this is exponential decay.1760

But one way to go about it is just to put it in the standard form, and then evaluate the value of the base.1765

All right, in Example 3, I have an exponential equation; and recall that, if the bases are the same,1773

then the exponents must be equal in order for the equation to be true.1779

The problem is that the bases aren't the same; so I need to get them to be the same.1787

Looking at these, they are both even; so I am going to try 2.1793

And we know that 2 cubed is 8; so let's start from there: 8 times 2 is 16; 16 times 2 is 32, so this gives me 16, 32...1796

times 2 is 64; times 2 is 128; so 2 to the third, fourth, fifth, sixth, seventh, equals 128.1817

If 2⁷ is 128, then 2^-7 = 1/128.1831

So, I have this written as a power of 2; let's look at the right.1839

I know that 2⁷ is 128; 128 times 2 is 256; 2⁷ times 2¹...1844

I add the exponents, so this is actually 2⁸ = 256.1856

Now, I can write this equation using the base of 2.1863

On the left, I am going to have 2^-7 raised to the 2x power, equals...1867

and then, on the left, 2⁸ times 3x - 1.1875

So, this is going to give me 2 to the -7 times 2x (is going to be -14x), equals 2 to the 8, times 3x - 1.1883

Now, I have it in this form, where the bases are the same, and I have an x and a y.1897

Therefore, -14x = 8(3x - 1); and then I am just left with a simple linear equation.1901

-14x = 24x - 8; I am going to add 14x to both sides to get 38x (I added 14 to both sides).1911

At the same time, I am going to add 8 to both sides to get it over here.1926

I could have done it the other way; then I would have had a negative and a negative, and then divided or multiplied by -1.1930

Anyway, I am going to go ahead and divide both sides by 38; and this is going to give me x = 8/38.1936

I can do a little simplification, because I have a common factor of 2.1949

I can cancel that out to get 2/19.1952

The hardest part for solving this was simply getting them into the same base.1956

And I was able to do that because 2^-7 is 1/128, and 2⁸ is 256.1961

Once I did that, it was simply a matter of putting the exponents equal and solving a linear equation.1967

Here we have an exponential inequality: again, the first step is to get the bases the same,1976

because I know that, if I have a base raised to a certain power, and it is less than that same base1981

raised to another power, that the relationship between these two exponents has to hold up.1992

Well, this time (last time I had some even numbers; I tried 2, hoping I could find a base there2001

that I could easily make them into a common base, and I did), since I see a 27, I am going to work with 3's.2008

So, I know that 3 cubed is 27; therefore, 3^-3 = 1/27.2015

243...let's look at that: if 3³ is 27, I multiply that times 3; that is going to give me 81.2028

If I multiply that by 3, fortunately, I am going to get 243.2037

So, this is 3³ times 3; that is 3⁴, times 3, so this gives me 3⁵ = 243.2044

Let's go ahead and work on this, then: rewrite this as, instead of 243...I am going to write it right here as 3⁵,2052

raised to 4x - 3, is less than...instead of 1/27, I am going to write this as 3^-3, all to 6x - 4.2063

So, this is going to give me 3 to the 5, times 4x - 3, is less than 3 to the -3 times 6x - 4.2074

Now, I have the same bases: I can just look at the relationship between the exponents:2087

that 5(4x - 3) is less than -3(6x - 4), and solve this linear equation.2092

This gives me 20x - 15 < -18x + 12; that is 38x - 15 (I added 18 to both sides) < 12.2101

So, 38x (I am going to add 15 to both sides to get this) < 27.2116

And then, I am just going to divide both sides by 38 to get x < 27/38.2123

Again, the most difficult step was just getting these to be written as the same base.2129

And I was able to do that by using 3 as the base.2135

And then, I had equivalent bases; I did a little simplifying, set them equal, and solved this linear inequality.2140

That concludes this lesson of Educator.com on exponential equations and inequalities; thanks for visiting!2150