Semester Final Review Spring 2026

Ajouté : 2026-05-23

[00:00:01]Hey guys, let's review for your semester exam. Now, this is going to be kind of a long one, but just stick with me.

[00:00:09]Hopefully, you guys can try a couple of them. It makes sense still. It may have been a while, but let's get back into what we need to know for our test. So starting off with transformations, remember that the negative out front is your reflection. Your A is your stretch or shrink. Your H is left or right. And your K is up and down. So on this one, it gives you your parent function. Okay?

[00:00:34]It tells you that there's a reflection.

[00:00:36]So we know that's the negative out front.

[00:00:43]It tells you that it shifts three right.

[00:00:45]Okay? Okay, so we know right and left is our h. So we know our h has to be three right is negative has to be -3 and then our reflection out front has to be a negative. Okay, so when we go to write the equation, we put a negative out front. We have 1 /x and your h always goes with your x. So minus 3.

[00:01:12]Now let's look at this next question. It is the square root function. It says it shifts right four. We know that's our h.

[00:01:20]Okay. One down. We know that's k.

[00:01:24]And then a shrink by 1/2. We know that's a.

[00:01:28]Okay. So, we can rewrite the equation.

[00:01:30]We have a 1/2 out front.

[00:01:33]We still have our square root of x for our four right. Remember, if it was going to the right, it's negative. So min -4 underneath and then one down. So it's minus one. Okay. Now let's try one more together. So we have 2 to the^ of x. It says there's a reflection over the x axis. So we know that's a negative out front and it shifts 2 units to the left. So we know that's our h.

[00:02:04]Okay. So our base is 2 to the^ of x. So we're going to have a negative out front.

[00:02:11]2 to the^ of x and then with that x we're going to have this uh + 2 for h.

[00:02:22]Okay. And these typically go in parenthesis.

[00:02:26]All right. So go ahead and pause the video and I want you guys to try these four, five, and six.

[00:02:36]Okay, let's go ahead and check your work here. This is what I got and I labeled all the things A, H, and K like we just did. Okay. So now let's look at this example here. It says describe the transformation. So notice it has an A, it has a K, it has a minus4. So it's only asking us to identify what the number is doing. So for this minus4, we know it's next to X, so it's our h value. And we know when h is negative.

[00:03:13]It's going to the right. So, this equation moved right four.

[00:03:20]Okay. So, I want you guys to do the same thing. Find the number that I gave you and then determine how it moved left or right. So, go ahead and push pause.

[00:03:32]Okay. And this is what I got. The three out front matches up with a. So, it's a stretch by three. The plus five is after your x. It's not in parentheses. So we know that's a k. So it goes up by five.

[00:03:46]So let's go ahead and look back at a polomial graph. And let's try to label some of the things. So let's start with our relative max. Remember a relative max means it's a high point but it might not be the highest point especially because the graph keeps on going up. And then our relative men, remember it's a low point but may not be the lowest point. But because this graph uh the end behavior goes up, we remember that this is our absolute men here.

[00:04:26]Okay. And then for our x and y intercepts, that's just where it touches the x- axis, where it touches the y- axis. So for our x intercepts, we have one here, one here, one here, one here, and then for our y intercepts, it's right here. It's the same as our x intercept at 0 0.

[00:04:50]Now, this equation is a little bit difficult to write an equation for. So let's just go ahead and identify what that parent function is. Okay. So it's an exponential function. So it has some base to the power of x. So f ofx equals some base b to the^ of x. Okay.

[00:05:16]Now moving on, let's look at this graph right here. So it says determine what the transformations would be. So we started at f. So, you're going to want to find one point on F. Let's say right here. We're going to find the same point on G right here. And then we're going to see how it moves. So, notice it goes 1 2 3 4 down.

[00:05:41]And it goes 1 2 3 4 to the right. So if we were going to write the equation f ofx oh I'm sorry g ofx g ofx equals okay so we have our f ofx and in there with our f ofx since it moves to the right four we're going to put minus4 and then because it moved down we're going to minus4 outside as well. Okay, so that just explains how our graph moved. It moved right four down four.

[00:06:25]All right, moving on to the next page.

[00:06:28]I would like for you guys to go through all the ones on this page. Try to describe all your transformations. Try to graph all of them. So, go ahead and push pause.

[00:06:42]Okay, so I identified all these transformations. I also graphed all of them. So let's start with this rational function. Remember, we have those rules and they're going to be on your test. So here's the rules for horizontal asmmptotes. So let's start there.

[00:06:58]Remember, you want to look at the degree on top and the degree on bottom. So you're looking at the biggest exponent with your x. Notice that we have a degree of one on bottom and zero on top.

[00:07:11]So our denominator is greater than our numerator which means that's example one. So we're going to say y equals z.

[00:07:24]Okay. And then recall for our vertical asmtote we want to set the bottom equal to zero. So for your vertical asmtote we want to say x - 2 = 0. We're going to plus 2 to both sides. So x = 2 is our asmtote.

[00:07:42]So we could draw that on our graph.

[00:07:46]Let's see we'll do yellow for our horizontal asmtote. That's right here.

[00:07:54]And then I'll do pink for our vertical asmtote which is on xals 2.

[00:08:01]Now remember, your domain and your range goes with your x's and your y's. So we're just going to say x does not equal 2 and y does not equal zero.

[00:08:14]Okay. So now looking at this question, how can we tell if it's growth or decay?

[00:08:20]So let's look at our table. Our y's are getting smaller. So if our y's are getting smaller, we know that's a decay. Okay.

[00:08:30]And then if you're writing your equation, recall that this type of function is y = a parentheses b to the power of x where your b is your growth or decay factor and your a is your y intercept.

[00:08:52]So looking here for your a, we want to look when x equals zero. So this is your A at one.

[00:09:02]And then for your B, you want to look at the pattern like how are we changing. So notice from here to here, we're multiplying by 1/2.

[00:09:12]From here to here, we're multiplying by 1/2. So that's your pattern. Your pattern is your b. Now to write our equation we have y = 1 parentheses 12 to the^ of x and then for this one we have a horizontal asmtote at y equals 0.

[00:09:48]Now let's move on to this next function.

[00:09:50]Remember these are square root functions. So they have an end point.

[00:09:55]That end point is right here on the graph. Okay. In order to find that end point, we just want to look at our transformations. How far it moved left or right, up or down. So because it moved right four up one, our end point is 4 comma 1.

[00:10:14]And notice it never touches your x or y axis. It's always greater than your x axis. It's always greater than your y.

[00:10:21]So there is none.

[00:10:26]There's no x intercept and there's no y intercept. And then for domain and range, we want to use the numbers from your endpoint. So, x is greater than or equal to 4.

[00:10:40]Y is greater than or equal to 1 because our graph is to the right and it's going up.

[00:10:48]Okay. So, another equation, our critical point, we're still looking at how far it moved left, right, up, or down. So, since it moved left one, it's -1, pos5.

[00:11:01]And for these functions, the domain was all real numbers and your range was all real numbers.

[00:11:08]Okay, let's go ahead and take a second to look at all of these answers.

[00:11:13]Okay, so for your critical points, remember you want to look at how far it moved left or right, up or down. For your domain and range, it's all real numbers. For a cubic function, for that square root function, it's going to the right, it's greater. Left, it's less than. It's going up, it's greater. If it's going down, it's less than. And you just want to use your critical points for those or your end point on that one.

[00:11:37]And then remember, it's growth if it's getting bigger, decay if it's getting smaller, and that a value is your y intercept. And then for any rational function, remember you have your set of rules. And when you're trying to draw your graph, you want to draw it based off of your um asmmptotes.

[00:11:57]Okay? So go ahead and pause the video.

[00:12:00]Try to describe all the transformations and graphs for 17 and 18.

[00:12:08]Hopefully you got what I got. The first one was a reflection over x and it moved up two and then the next one just moved up five. Okay. And then hopefully your graphs look like my graphs. All right.

[00:12:23]So now going back to your critical point, it moved up two. and it didn't move left or right. So we have 0 comma 2. And then for your domain and range, it's the same thing as the cubic functions which was all real numbers.

[00:12:41]And then looking at this graph, your asmtote is horizontal.

[00:12:48]Okay? So we have a horizontal asmtote right here. This comes for these functions. It comes from your k value. So we just want to do y = k.

[00:13:01]And in our case, our k is 5. So y = 5 is our answer for that one. Domain is all real numbers, but your range is y is greater than 5 because it approaches but never touches five.

[00:13:18]So those are what that looks like. Now, here are the rules for log. And I would pause this, write it down, maybe put it on your note card, and then try to do these questions. So go ahead and pause it, write these rules down, and then do all of the questions for 19 and 20.

[00:13:42]So go ahead and check your work for 19 and 20. This is what I got.

[00:13:46]Okay, notice that on those expanded forms, you have two rules. Okay, you have the um the quotient rule and then you also have the power rule and then on the bottom one you have the power rule and you also have the product rule.

[00:14:03]Okay, so you want to make sure you write it out for each one of the different rules. Now let's go ahead and look down here. So we're going to use we're going to try to solve for x using logs and exponentials.

[00:14:18]Um, when we're solving for x, you want x.

[00:14:22]You want it to be x equals. We want to isolate for x. So, looking around, find where your x is. That's where you want to get it isolated. You want to get it by itself.

[00:14:34]Okay? So, the first three are pretty straightforward. So, go ahead and try those first and then I'll explain the next three. Go ahead and pause it.

[00:14:46]All right. Here's what I got. Hopefully yours looks similar to mine.

[00:14:53]Okay, so now let's go ahead and take a look at number 24.

[00:15:00]Okay, so we notice that our x is in the exponential up there. So we need to isolate this first. So that means we have to move a minus 5 and a six. And anytime that number is in front of something with no plus, minus or division sign, then you know that's multiplication. So we're going to do the opposite of subtraction, the opposite of multiplication. So I'm going to plus five to both sides and I get 25 = 6 to x. Then I'm going to divide by 6.

[00:15:39]Okay, so I have 25 / 6 = 2 ^ x. Now I have my exponential by itself and I can rewrite it as a log.

[00:15:51]Okay, so we have log base 2 of 25 / 6 = x. And then you now that we have our x by itself, it says x equals We can just plug this into our calculator and get our answer.

[00:16:13]Now, let's look at number 25.

[00:16:16]Notice number 25 has something similar on both sides. And that's the same base.

[00:16:22]And because it has the same base on both sides, we can just ignore our bases and set our exponents equal to each other.

[00:16:30]So, we have 2x - 1 = 7. And then you just solve like we normally do. + one to both sides.

[00:16:39]2x = 8 and then divide by 2.

[00:16:46]So, x = 4.

[00:16:50]Now, we have our x equals by itself and then we're finished solving that one.

[00:16:54]Now, let's take a look at number 26.

[00:16:57]Okay, we have x's in both exponentials.

[00:17:00]So, we need to get the same base. And remember that 81 can break up into two 9ines. 9 * 9. So we can rewrite 81 as 9^ 2ar. So we have 9 ^ of 3x = 9^2 x - 4. When we have it like this, we want to go ahead and distribute the exponents. That's our exponents rule. So we have 9 ^ 3x = 9 ^ 2x - 8.

[00:17:36]Make sure you get it all the way to the 4. Another way to do that is to do the box method.

[00:17:45]2x - 8. Okay. Now that our bases are the same, we can go ahead and ignore them.

[00:17:52]And we have 3x = 2x - 8. I'm going to minus 2x to both sides to get our x's all the way on the left and I get x=8.

[00:18:08]Okay, so that's it for this page. Let's move on to this one. So recall for your exponentials your asmtote is x= k and for logs your asmtote is y= h. It's positive if it moved to the right, negative if it moved to the left, and it's zero when there's no h or k. So, go ahead and try to do 27 through 30. Go ahead and push pause.

[00:18:38]Hopefully, you got what I got. y = 6, y = 0, x= -2, x= 0. And then recall for these b's, that's your growth or decay factor. Those are those exponential equations. Okay? So if it is growing, we want our graph to go up. It's increasing from left to right. And when it's a fraction, it's decaying and it's going down from left to right.

[00:19:11]Okay? So I want you guys to go ahead and try to identify um if it's a growth or decay factor. So remember, you're looking for your B and you're trying to see if it's greater than one or if it's between zero and one. If you're not sure, you can graph it. See if it's going up from left to right or down from left to right. So go ahead and push pause. Oh, and actually go ahead and answer number 33, too.

[00:19:34]That's pretty straightforward. It's just what are the inverse functions? Try to see what you remember.

[00:19:41]Okay, here's what I got. I got growth, growth, decay. Then the inverse of a log is a exponential. Inverse of a cube root is a cubic. And the in the inverse of a quadratic is the square root or radical.

[00:19:55]So now let's solve these equations. So they h we have some square roots and we have some cube roots, but it's the same thing. You want to isolate the radical, square or cube both sides, which is the opposite. isolate for x and then plug that into your original equation to see if it's real uh true or false. Okay, so why don't you guys try step one, pause the video, try step one for all three of those questions.

[00:20:28]Okay, hopefully you guys got um you added five to both sides and divided by three to get 35 by itself and then you just minus one to both sides to get the cube root by itself. So, let's go ahead and work out the rest of the problem together. So, we want to square or cube both sides. So, this first one, we're going to go ahead and square it because it's a square root.

[00:20:55]So, we have 2x - 7 = 9. We're going to plus 7 to both sides. 2x = 16 and divide by 2.

[00:21:09]So, x = 8 is your final answer on that one.

[00:21:15]Uh, let's look at this one. So, we're going to square both sides again.

[00:21:22]2^2 is 4. So we have 8 x + 1 = 4 - 1 to both sides.

[00:21:32]8 x = 3 / 8.

[00:21:38]So x = 3/8.

[00:21:44]Okay. And let's do this last one. Notice it's a cube root. So, I'm going to go ahead and cube both sides.

[00:21:55]Okay. And I get 2x = 64.

[00:22:00]Divide by 2.

[00:22:02]x = 32.

[00:22:08]Hopefully, your algebra has gotten better over the year and those feel pretty straightforward. So, finding the inverse. Remember, these are the steps.

[00:22:17]You want to rename the function y, switch your x and y, then isolate for y, then you're going to rename the function. So try step one and two for 37 and 38. Just rename the function y, switch your x and y, and we'll go from there.

[00:22:35]Okay, hopefully you got what I got here.

[00:22:37]So notice I instead of writing f ofx, I wrote y. Then I switched my x's and y's.

[00:22:44]Now we're going to solve it. So the cube root is here. So we needed the opposite of a cube root to get rid of it. So we're going to cube both sides. So we have x ^ 3 = y - 2. We're still isolating for y, so I'm just going to add two to both sides.

[00:23:03]So y inverse = x cubed + 2.

[00:23:10]Now let's check this question.

[00:23:13]So notice we have this log. We want to get that by itself first. So in order to get this by itself, we need to move this minus one.

[00:23:23]So I'm going to add one to both sides and I get x + 1 = log base 2 y + 5.

[00:23:35]Okay. And then we're going to do 2 ^ x + 1 = y + 5. Just rewrite it. and then minus five to both sides. Okay, so um we're gonna say f inverse x = 2 x + 1 - 5.

[00:23:58]Okay, so those are inverses.

[00:24:02]Let's work on uh these next two questions 39 and 40. Try to find your x and y intercepts. Remember, you can graph it. You can trace your graph. Um, you can plug in zero for your x inter uh for your x's. Plug in zeros for your y's. Find it that way. Find your y intercept by making a table looking for x equals 0. Whichever way makes most sense to you at this point.

[00:24:26]Okay. So, here's what I got. Now, let's look at our domain and our range for these two questions. So 2 to the^ of x remember that's an exponential and it is growing it's going up.

[00:24:44]So those have all real numbers for your domain and for your range are y is greater than zero because that's where the asmtote is.

[00:24:56]And then for logs it's going to be the opposite because they're inverses of each other. And our x is going to be equal to um oh I'm sorry our x has to be greater than how far it moved. So it moved right one.

[00:25:18]So it has to be x is greater than one.

[00:25:21]That comes from your h value there.

[00:25:24]Now that's it for this page.

[00:25:34]Let's go on to this next question. So recall critical points is your h and k.

[00:25:40]So go ahead and try these. Remember if it moved left your h is going to be opposite. It'll be negative. And if it moved right your h is going to be positive.

[00:25:50]So I got -5 comma 0 -3a -2 1 comma 2.

[00:25:57]And then looking at these rational functions, I want you guys to try to find your x intercepts and your horizontal asmmptotes. Remember, here's your rules for horizontal asmmptotes.

[00:26:09]So, pause the video and try that.

[00:26:16]Okay. So, here's what I got. So, on that first one, they were equal to each other. So we're looking at the number in front of x^2 in the on the top and bottom which is -3 over 1. Okay. So let's go on to the next page to find your holes and your asmtotes in your domain. So for this question we have to factor the top and bottom. The top is just one term so it doesn't need to be factored but that bottom if we recall that's our difference of squares. So to factor it, this is going to be equal to -3x^2 over x - 2 x + 2.

[00:26:59]For your holes, you look for something on top and bottom that cancels. But because nothing cancels, you know there's no holes.

[00:27:07]All of these rules are given to you on the test. So if you don't remember, you don't need to write them down. Then for your vertical asmtote you set the remaining factors in the denominator equal to zero. So we want to do x - 2 = 0 and x + 2 = 0. So that gives us x= pos2 and x= -2 for your asmtotes. Then your domain is x does not equal your holes or your asmtotes. So x does not equal 2. x does not equal -2.

[00:27:41]That was probably a little bit fast, but let's go ahead and try this next one.

[00:27:47]Okay, notice the top is already factored. The bottom is going to be your diamond method. So, we want to label our A, B, and C. So, A= 2, B = 3, and C = -2. So in order to do our diamond, we have a * c on top, b on bottom. So we have -4 on top, three on bottom. And then we're looking for two numbers that multiply to get -4 and add to get positive3.

[00:28:22]So we have 2 x on top and then I know four and one can make three. Okay. and we need a positive three. So I'm going to put pos4 and a1.

[00:28:39]And then the only thing we have left to do is simplify.

[00:28:46]So notice this 2x over 4 can simplify to 1/2. So this one, let's go ahead and erase these. We don't need those anymore. This one is going to simplify to 1 x over 4. So to write our parentheses, we have x + 4 and 2x - 1.

[00:29:10]Now let's rewrite our function. We have -2 x + 4 over x + 4 2x - 1.

[00:29:28]Notice we have an x+ 4 on top and bottom. So that's going to be our hole.

[00:29:34]So for our holes, we have x + 4 = 0. So x = -4 is our hole. And then for our vertical asmtote, we're going to take what's left, which is the 2x - 1. So for our vertical asmtote we're going to say 2x -1 = 0 + one to both sides 2x = 1 / 2 x = 12.

[00:30:06]Okay. So um remember for our domain it's just x does not equal your holes and x does not equal your asmtote.

[00:30:24]That was a that was a hard unit for us.

[00:30:27]But moving on from that one let's go ahead and identify some things. So we can identify our vertical asmtote on here that's just right in the middle.

[00:30:37]Our horizontal asmtote that looks like it's right about here at one.

[00:30:47]And our hole is right here.

[00:30:53]Our x intercept is where it touches the x axis. And then our y intercept is where it touches the y ais.

[00:31:03]And then recall it's just x does not equal your holes or vertical asmtote which was at -1 and -3. So x does not equal -1 or -3 and y does not equal 1.

[00:31:18]Okay. So let's go ahead and take a look at some word problems. Remember it's 1 over person A + 1 over person B = 1 over the time together. So Sarah can do the job in 4 hours. John can do the job in 5 hours. and Mike can do the job in 6 and 1/2 hours. And then what would it take for them to do it together? So we have 1 over 4 + 1 over 5 + 1 over 6.5 = 1 over t, which is our unknown.

[00:31:53]And then remember to solve it, you're going to multiply each fraction by the common denominator. So multiply everything times 4 * 5 * 6.5 * t.

[00:32:07]Okay. So then you just see what cancels.

[00:32:11]So on that first one the fours cancel.

[00:32:13]So we have 1 over 5 * I'm sorry we have 5 * 6.5 t + 4 * 6.5 t + 4 * 5 t = 4 * 5 * 6.5 Okay, so you just add up all those numbers and that's your that's what you get.

[00:32:51]For this next one, you want to identify your P, N, R, and T. So, we started with $80.

[00:33:02]That's your principal. Your rate is 5%.

[00:33:05]So, 05.

[00:33:07]And the number of times is 12 because it's monthly. And it's asking what it would be after five years. So you're just going to plug all that information into the equation and plug it into your calculator.

[00:33:23]I am not going to do that part.

[00:33:26]Let's go ahead and look at number 47. So for this one, you want to add your list in spreadsheets. And in order to do a power regression, you press menu 419.

[00:33:36]Make sure you select that x and y in the table and then plug in um your a and b into the equation.

[00:33:43]So go ahead and check your work with me.

[00:33:47]This is what I got.

[00:33:50]Hopefully you got the same thing. Notice our exponent is 1/2 and or.5 and anytime you have x to the 1/2 it's the square root. So you can write it either way.

[00:34:01]Now let's look at this one. These are rational functions. In order to graph them, it's best to graph your vertical and horizontal asmmptotes first. So your vertical asmtote on this first one is at four because it moved to the right four.

[00:34:20]And this one is at zero because it didn't move left or right.

[00:34:24]And then for your vertical asmmptotes, this first one is at zero because it didn't move up or down. The second one is at pos4 because it did move up or down.

[00:34:35]Okay. And then just go ahead and draw in your lines.

[00:34:42]Okay. So, hopefully this is what yours looks like. I know this is a lot of information. Um, do your very best to study, practice questions, pause the video, make sure you're actually trying to work some of these questions out so that when it comes time for the test, you actually know how to do it. All right, good luck, guys.