Solving polynomial functions using their graphs
About this video
Using the graph of a polynomial f, the video finds the x-values where f(x) = 0 by reading where the graph meets the x-axis (x = -2, -1.5 and 3). It then finds where f(x) = 15 by drawing the line y = 15 and estimating the three x-values where the graph meets it, giving the integers or halves they lie between.
You'll be able to
- read the solutions of f(x) = 0 from a graph as its x-intercepts.
- solve f(x) = k from a graph by finding where it meets the line y = k.
- estimate a solution between two values when it is not at a gridline.
Key words
- x-intercept
- A point where a graph crosses or touches the x-axis, so the output is 0.
- zero of a function
- An x-value that makes f(x) equal to 0.
- approximate
- To give a close estimate, such as 'between -2.5 and -3', when the exact value can't be read.
Transcript
- [Instructor] So I have the graph of a function f here. And the first thing I want to do is I wanna figure out all of the x-values where the function intersects the x-axis. Or another way to think about it is, what are the x-values where f of x is equal to zero?
So pause this video and see if you can pick those out. Okay, so when f of x is equal to zero, that means we are intersecting the x-axis, so that's there, that is there, and then that is there. And we can see what these points are. It's a little bit small on the screen, but we can see that that happens at x equals negative 2.
So we could say, for example, that f of negative 2 is equal to zero. We can see that, let's see, this is, if that's negative 2, this is negative 1, this is negative 1 1/2. So F of negative 1.5 is equal to zero. That's this point there. And then this last one over here, that looks like 3 x equals 3, so f of 3 is equal to zero.
Fair enough. Now let me ask you another question. Well, we're gonna think about at what x-values does the function equal 15? Well, when you look at it that way, if you say 15, right over here, that's the function, that's y equals 15. And it looks like it's happening right over here, right over here, and right over here.
So it's not happening exactly at any integer values of x, for example, this looks like it's happening at a point that is not quite at negative 3. It's between negative 2 1/2 and negative 3. It's much closer to negative 3. So I could write here it's between negative 2.5 and 3, or if we...
or let me be very careful. That's between, oh yeah, negative 2.5, negative 2.5 and 3. If we said what integer is between, we could say it's between negative 2 and negative 3. So between negative 2.5 and negative 3, I should say, we need be very careful. Or if we wanna say between what integers, we could say it's between negative 2 and negative 3.
Similarly, this point right over here, it looks like it's between x equals... it's happening between X equals 0 and x equals negative 1/2. If we want to think about integers, we could say it's between zero and negative 1. And then this point right over here, it looks...
integers is happening between x equals 2 and x equals 3, that the function takes on the value of 15. And if you wanna get a little bit more precise and you're willing to go beyond integers, it's between 2 1/2 and 3. So I'm just doing this so you can practice just thinking about when does a function take on certain values.
Sometimes you can read it pretty clearly and you can get the exact answer. When you're looking at a graph, ideally, they make it very clear that the graph is intersecting the axis at those points, maybe by drawing some dots, and that sometimes you might have to approximate, like we just said, when does this function equal 15?
Well, I don't have the exact x-value just by looking at this, but I can tell you roughly where it is. It's between negative 2 1/2 and negative 3, and it's actually much closer to negative 3.
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