2026 AP Precalculus Free Response #2

追加: 2026-05-15

[00:00:00]All right, let's look at the F FRQ number two of the AP pre-calculus 2026 exam. So, a person purchased a car at the end of 2019. The car's value decreases over time. At the end of 2020, the car is valued at this. At the end of the year 25, the car is valued at this.

[00:00:14]Okay. The value of the car being modeled by this function exponential decay function is the value of the car and number using the given data to write two equations that can be used to find the values of the constants a and b. So, we have two points. We know at 1 it's equal to 27.2. So we know 27.2 that's the value is equal to a b to the 1.

[00:00:36]And then we also know when we plug in t= 6, it's going to equal 14.8. 14.8 is equal to a b to the 6. Right? Like this.

[00:00:46]So that's two equations there. So two equations, two unknowns. We can solve for them. Um, easiest way to do this, you could do this just strictly on the calculator, but I like to divide these two. So, [snorts] AB to the 6 over AB is equal to 14.8 over 27.2.

[00:01:05]That's going to cancel. B 6 over B is B to the 5th. So, B is going to be the fifth root of 14.8 over 27.2.

[00:01:16]So, we're going to write this as 14.8 / 27.2 2 raised to the 1/5 power. It's how I always do that radical form, right?

[00:01:25]That's fifth root. 0.885 is for b.

[00:01:31]And then you just plug back into here. I know a * b has to equal 27.2. So a is equal to 27.2 / 0.885.

[00:01:40]And we just do a divided by don't use the rounded value. Try to get that whole value there. Take that and copy it in there. Uh, oops. I don't know why I said a. 27.2 over that. Then a is going to equal 30.721.

[00:01:58]Like that. Using the given data to find the average rate of change of the value.

[00:02:03]So let's just go ahead and put this as b equals and we'll make this a equals.

[00:02:09]Just put this over b. It's one way you can do it just to have the numbers in there and a and b. And then we'll just say v of t is equal to a b to the t.

[00:02:19]Right? So that gives us a function. Uh let's delete from question one all that stuff. Okay. Um find the average rate of change. Average rate of change is a secant line slope. Slope between two points. Um so it's going to be v of 6 minus v of 1 over 6 - 1. You're basically taking the two y values at 1 and 6 and then you're calculating that.

[00:02:43]And now that we have this notation, we can do V of 6 - V of 1 over 6 - 1. Or you could just plug in the values because they already told you what V of 6 and V of 1 is. This minus this, I get -2.48. Let's make sure we put in our units, the correct units. Um, so in in thousands of dollars per year, so thousands dollars per year, however you want to write that is fine.

[00:03:13]Okay. Number two, use the average rate of change to estimate the value of the car in thousands of dollars at t equals 3. Show the work that leads to your answer. So, if we're going to use that average rate of change, there's a couple ways you can do this. In calculus, we always teach it as like a use a tangent line approximation. Um, but basically, you're going to start at V of one.

[00:03:34]This is one way to do it. You can also do a tangent line. I'll kind of show you both ways if you want. You're going to start at this point. you're going to say like, well, I'm going to then do the this rate of change and we've gone over two years because this is per year. So, we're going to multiply by two years.

[00:03:49]Why two years? Because we're asking about 3 years, which is 2 years after the one year. So, it's the that initial value plus 2 years times that rate of change. So, that would be one way to look at it minus 2.48 * 2 and that's going to equal 22.24 thousands of dollars.

[00:04:12]like that. The other way to do it is a more calculus way is like um I'm just showing this because if you take AP calculus next year um we always use it as a point slope form.

[00:04:26]This is equation of a line. Any equation of a line can be done this way. Right?

[00:04:30]So the point here is y minus we'll do the v of one value. We'll say v of one.

[00:04:35]That's the value there. The y value is equal to the slope. And the slope is that average rate of change x -1 or t minus one. In this case, t is the variable instead of x, right? And so then y is going to be v of one. Move this to the other side. Minus 2.48 t minus one. And then you just plug in t is equal to to a two here or three. And notice 3 - 2 is equal to 1 or sorry 3 - 1 is equal to two times that is going to it gives you the exact same expression.

[00:05:06]So either way you want to think about it is the exact same thing but point slope form is like my recommendation just because that is very commonly what we'll do in calculus. All right. Average rate of change can be used to determine the secant line for the graph from here. Let a of t represent the estimate of the value of the car in thousands of dollars using the seeant line for a of three found in part b. It can be shown that a of three um let me see about this real quick.

[00:05:32][snorts] Okay they're just reexplaining in here. I had to just parse it out for a second. They're just saying that our estimate was using a seeant line. It's like a line that goes through two points. It can be shown that a of three is greater than V of3. In general, a of t is for all t use a seeant line of graph to explain why this is true. Okay.

[00:05:49]Um it's true because um I I just remember in calculus it's concave up. So the line the seeant line if you consider the seeant line here and let's let's let's draw that seeant line y is equal to v of 1 - 2.48 4 8 x - one. This is the secant line here. And basically what's happening is the secant line between these two points one and six.

[00:06:12]Right? So we've drawn a line in between.

[00:06:14]It's above it. It has to do with the fact that it's like the it's concave up.

[00:06:18]So I'm going to put both of them because I don't exactly remember the justifications that are valid for this.

[00:06:24]Um it is the seeant line and the graph to explain why this true. Well, the the seeant line is above v of t because um uh v of t is concave up.

[00:06:43]Or another way to think of it is the the the rate of change decreases.

[00:06:55]Uh so the um yeah so the rate of change de uh decreases or I guess it technically increases because it's getting less negative rate of change increases.

[00:07:13]So the um the the average one is an overestimate.

[00:07:26]Okay.

[00:07:29]All right. So for part C, when the car of the val when the value of the car reaches $2,000, Carter plans to donate it to an auto mechanic school. As a result, the car will immediately lose all of its value. explain how this information can be used to determine the domain limitation for the model V. Okay, so basically um notice that this V assumes like if we look at the the graph of V, it assumes that we're just going to decay forever.

[00:07:57]But once it's gets to two, which is $2,000, we're going to stop right here. Okay, so basically at this point it the model says it would continue decreasing in value, but they're telling you here that it's going to just it's immediately going to go to zero. So prior to after this point it's invalid because after this point it should just drop to zero.

[00:08:20]So the v of t is only valid for uh v for v of t greater than equal to to two.

[00:08:31]So when v of t is equal to two then the function no longer is no longer is no longer representative of the value of the car which is zero at that point. $0 after that point.

[00:09:04]So it just says to explain, you don't actually have to find it, but we did find it. The domain would be from 0 to 22.444 444 years as an example, but they just said to explain, not to actually do