Interpreting equations graphically | Mathematics III | High School Math | Khan Academy

Let F of x = 3x - 5 and g of x = x^3 - 4x^2 + x + 6.

The graphs of y = F of x and y = G of x are shown below, and we see them right over here. This y = F of x is in, that is, in that purplish color. Let me see if I can get that same purplish color so that is in that. So let me underline it now.

So F of x is in that purplish color, so y = F of x that's right over here in purple. And then y = G of x, that is in blue, so y = G of x, G of x is defined right over there. y = G of x, well that is graphed in blue, and we see that they intersect at the point (a, b).

So there's a couple of ways to think about this. We could say that when x is equal to a, F of x and G of x equal each other, or we could say F of a, and this is coming from this point of intersection.

Let me draw a little arrow here, so or big arrow, that point of intersection lets us know that F of a is equal to G of a, which is equal to b. G of a, which is equal to b. They both, if you input a into the function F, you're going to get b.

If you input a into the function G, you're going to get b. And so the point (a, b) is on both graphs, both y = G of x and y = F of x.

From here, you can make some interesting statements. For example, you could just say, well, what is F of a? F of a is 3 to the a power - 5 is going to be equal to what's G of a. G of a is a^3 - 4a^2 + a + 6.

So, you could say this, and that would be equal to that. Either of those would be equal to b.

All right, I think we've analyzed that a good bit, so now let's actually answer their questions. Normally, I'd suggest that you look at the questions before you actually try to solve, but I just wanted to do this just to really squeeze out as much as we could out of the information they gave us.

So they tell us the value x = a is a solution to which of the following equations. Select all that apply. So this first one is 3x - 5 is equal to b. Well, we already know that.

We already know that 3 to the a - 5 is going to be equal to b, is going to be equal to b. This over here, this, this is equivalent to saying F of x is equal to b, and we know F of x equal b when x equals a.

When x equals a, F of x is equal to b. This expression is equal to b, so we know that that first one is true. Now, the second one is just saying F of x is equal to G of x.

Well, we know that when x is equal to a, F of x is equal to G of x, that F of a is equal to G of a because, as a reminder, this right over here is our definition for F of x, and this over here is our definition of G of x.

So this is just saying F of x equals G of x. When does that happen? Well, that happens when x equals a. We already saw it up here F of a is equal to G of a; they both equal b.

And so both of these are going to be equal to each other when x is equal to a. So I will check that one as well.