Hey, guys. Welcome to Algebra 1. Today's lesson's going to focus on finding X- and Y-intercepts of linear functions. Your knowledge of evaluating expressions in the coordinate plane is going to come in handy for this lesson. You ready to get going? Let's start.

(Describer) She uses a stylus on a screen.

Okay, before we start digging into finding the X- and Y-intercepts of linear functions, let's throw back to graphing linear equations. Here we have the graph of Y equals 2X minus 1. And knowing what we do about relations in functions, I know that I could apply the vertical line test here and that I could tell

(Describer) She draws a vertical line.

that this relation is a function.

(Describer) Lines cross once.

Once I know that, I can use what's called function notation, because now I know that this relation is a function of X.

(Describer) She writes.

For every input there's only one output for this relation, right? Now I know that it is a function. Once I know that, I can replace Y with F of X. Let me erase my vertical line here so we don't get too crowded, okay. Once I know that my relation is a function of X, I can, instead of using Y, say that it's a function of X, so that F of X equals 2X minus 1. That's how we read that, "F of X," just shorthand for "function of X." Keep that in mind, and let's look at this graph here. Knowing what we do about linear equations already, I know how to investigate and analyze this graph and determine where my X-intercept is and where my Y-intercept is. I know my Y-intercept is the point where my graph crosses the Y-axis. In this case, I'd have a Y-intercept of negative 1. I know my X-intercept's where it crosses the X-axis. In this case, it appears to be half. In this lesson we'll dig into how to get to those values just given algebraic representation of a function, so just given "F of X equals" and then an expression. Then we'll throw back to the graphs to make sure you get the connection. All right, let's get going here. So given, function of X, or given F of X equals 5X minus 15, we're going to find the X- and the Y-intercepts. So we know, when we have an X-intercept, that our Y value is always 0. Let's throw back to that graph, so I can show you exactly what I mean. So for this value here-- get the pen-- I know it's an X-intercept, and that is the value one-half, 0. I know if I have an X-intercept, the Y value has to be 0 because I'm on the X-axis. And my Y-intercept, right here, this one is at (0, negative 1). That makes sense, because any time I have a Y-intercept, I know that the X value has to be 0 because I'm on the Y-axis-- keep that in mind, and we'll look back at this problem. Know that, if you have an X-intercept-- I'll do a little shorthand work over here-- for my X-intercept, I know that my ordered pair will always take the form that there's some number for X, but Y must be 0. Then for my Y-intercept, I know those ordered pairs always take the form that I have 0 for Y-- or 0 for X, and my Y value is some number, but X has to be 0, right? So we're going to use that information to find our X and Y intercepts algebraically. So first, let's get the X-intercept. Going to abbreviate that here. For my X-intercept, I know that Y must be 0. Because we know that F of X is another way to know that that's Y-- when we see F of X, we can just say, "That's Y." I'll replace F of X with 0. I'm going to say 0 equals 5X minus 15, because if I have an X-intercept, the Y value must be 0. Then I'll just solve this equation for X, and I'll have the X value for my X-intercept. So let's get this solved. We know we want to isolate the X. So let's start out by adding 15 to both sides. So we know that's going to cancel out there. We'll have 15 equals 5X. Let's scroll a bit, get some more space. We know our last step here would be to divide by 5. So 3 equals X. That tells me that my function will cross the X-axis at 3. I can represent this as an ordered pair by saying, well, then the X-intercept is (3, 0). That's the number, that's the value of X in this case, okay? Let's scroll back up. Now, to get my Y-intercept, like we said, we know that X has to be 0, right? In this case, instead of replacing F of X with 0, we're going to replace X with 0, all right? Let me scroll down here just to get some more space. It's going to get a little crowded if I go right beside there. All righty. So we're going to find our Y-intercept. And we know that our function was F of X equals 5X minus 15. And we know that for our Y-intercept... that we always have 0 for X, and then some numerical value for Y. What we're going to do to find our X-intercept-- sorry, our Y-intercept-- is replace X with a 0. We're going to do what's called find F of 0, because we're going to replace that X value with 0. We'll say, all right. Well, then 5 times 0 minus 15. 5 times 0, that's just 0. So 0 minus 15, which is negative 15. So we know here that our Y-intercept is located at (0, negative 15), which we kind of probably knew a little bit intuitively because this is also an equation written in slope intercept form, right? And we know for slope intercept form, that our Y-intercept is always B-- or always that constant term on the end, right? You could also look at it that way. Just to get comfortable with using function notation, I'll substitute that value of 0 in there for X every time, just to get comfortable with that process, because it'll be handy a little later. Let's move on to another example. Here we have the function, one-third X plus 4. And you want to find the X- and Y-intercepts. Okay, let's get the X-intercept first. Doesn't matter which one, so I'll go in the order that it was presented to me. I always write down, just to keep it straight. I know that for my X-intercepts that I'll have some value for X, but that Y is always 0. So if my function is F of X equals one-third X plus 4, I need to replace that F of X, because I know that it represents Y, I want to replace that with 0. So I will have 0 equals one-third X plus 4. Then I'm just going to solve this equation for X. Get a little more space. So I'll subtract 4 from both sides, right? So this is going to cancel here. Zero minus 4, that's negative 4 equals one-third X. Let's get a little more space here. Now I need to get rid of this fraction. So I know I can multiply both sides by 3. 3's will cancel there. 3 times negative 4, that's negative 12 equals-- and then I'm just left with 1X or just X. I know here, my X-intercept is at negative 12. Or I could represent it

by the ordered pair (negative 12, 0). I know that that is the location of my X-intercept, okay? One thing's checked off the list. We found half of the problem here, half of the answer. Now let's turn and get the Y-intercept. And I may have given myself enough space. I don't want it to get too crowded. I'll come down underneath here. Okay, so now we'll find the Y-intercept. And we know for the Y-intercept, because I always like to give myself a little hint, I know that X will always be 0, but I'll have some numerical value for Y. Let's scroll back up so we can get that function. So our function of X is F of X equals one-third X plus 4. So F of X equals one-third X plus 4. We know for Y-intercepts that X is always 0, so I'm going to replace the X in my problem with 0. I'll evaluate that and see what I get there. So F of 0 now-- because I'm replacing X with 0-- equals one-third times 0 plus 4. Let's get some more space. Okay, we know that one-third times 0 is just 0. So that's 0 plus 4, which is 4. Which, like I said, we knew that already, because it's in slope intercept form, but there is a reason why you should do this process. It's going to come in handy really soon down the line. So I know that my Y-intercept is at 4. So I can write that as an ordered pair (0, 4). And there we go, you're all done. You've got both your X-intercept and your Y-intercept for this function. All right, let's keep moving. It is time for you to try one. So press pause and work your way through this problem. Go ahead and find the X- and Y-intercepts. When you're ready to check your answers, press play.

(Describer) Titles: Given: f of x equals 8x minus 3. Find the x and y intercepts.

(female narrator) Given: F of X equals 8X minus 3. Find the X- and Y-intercepts. Okay, let's see how you did here. Switch to my pointer tool to reveal these answers. So the X-intercept for this function is at 3/8ths, 0, and the Y-intercept for this function was at (0, negative 3). To see how I got that, stay with me, and I'll show you my work. So first, to find the X-intercept... I'll write it in the middle this time. There we go. I always make myself a little note that I know for my X-intercept I have some value for X, but Y is 0. So I'm going to replace F of X, which I know is basically like Y, with 0. So 0 equals 8X minus 3. Let's scroll down here. Now I just need to solve this equation. I'll add 3 to both sides. So that cancels out there. And I have 3 equals 8X. Scroll down a little more. Divide both sides by 8, and you have 3/8ths equals X. Then to write that as a ordered pair, you could write (3/8ths, 0). Then you would have the location of your X-intercept, okay? Now, to get the Y-intercept, we want to let X be 0, right? Let's get some room. All right, so we're going to get our Y-intercept now. For the Y-intercept, we know that X is always 0 and then Y is some numerical value. Let's scroll back up to get that function. So F of X equals 8X minus 3. So F of X equals 8X minus 3. We're going to find F of 0, because we're going to replace that X with 0, okay? So 8 times 0 minus 3. 8 times 0 is 0. 0 minus 3, that's negative 3. So the location of our Y-intercept, here, if I wrote it as an ordered pair, would be 0, negative 3. All right, good work there. Let's keep moving. We'll bring it back to the graph to make sure you got that connection between what you're doing algebraically and then what it looks like on the coordinate plane, okay? We know for X-intercepts, that we-- it's easier to tell on the coordinate plane that your X is 0 because you're looking for the point where your graph is going to cross the X-axis. I see that point's right here. I can say my X-intercept is located at (2, 0). I've got it that quickly. I know for my Y-intercept, I know that X is 0. I'm looking for the point where my graph crosses the Y-axis. I'd look right here, in this case. It looks like I'm at one, two, three, four, five, six. One, two, three, yep. So (0, negative 6). And there's the location of my Y-intercept, there. So still a function, but now we're looking at it represented by its graph. We can still get the X- and the Y-intercepts. Let's keep going.

(Describer) She switches to another graph.

Okay, so let's get the X- and the Y-intercepts here. So X-intercept, looking for the point where your graph crosses-- crosses the X-axis. Here we go, right here.

And that is the location (3, 0). Y-intercept, I'm looking for where my graph crosses the Y-axis. Let's see, that's way up here. One, two, three, four, five, six, seven, eight, nine. So at-- get my pen here-- that's (0, 9). There's our Y-intercept. Then at (3, 0), there was our X-intercept, okay? All right, let's keep moving. It's time for you to try one. Press pause and take a minute and locate the X- and the Y-intercepts of this linear function. When you're ready to check your answer, press play.

(Describer) The linear function crosses the x-axis at negative 2. It crosses the y-axis at negative 8.

(female announcer) The linear function crosses the X axis at negative 2. It crosses the Y axis at negative 8. All right, let's see how you did. It looks like our X-intercept's located right here... and we are at (negative 2, 0). Our Y-intercept--we're kind of down low--let's count. Negative 1, two, three, four, five, six, seven, eight. So our Y-intercept is at (0, negative 8). Okay. All right, good job on those.

(Describer) Next is a set of threee ordered pairs: 2, 28; 0, 14; and negative 2, 0.

(female announcer) Next is a set of three ordered pairs:

(2, 28), (0, 14), and (negative 2, 0). Now we're looking at-- again, we have a function-- but we're looking at it by a list of some of its ordered pairs. We want to find, what's our X-intercept, what's our Y-intercept? In this case, it's important that you remember that shorthand little trick I wrote down that when you're looking for your X-intercept, you're looking for some numerical value for X; the Y must be 0, right? And when you're looking for your Y-intercept, X must be 0 and you're looking for some numerical value for Y, right? It looks like here, I see I have (0, 14), and I have (negative 2, 0). So for my X-intercept, I know Y has to be 0. So there we go-- that's my X-intercept. So my X-intercept is at (negative 2, 0). I know for my Y-intercept, X must be 0. So there we go, our middle point, right there. So for our Y-intercept, it's located at (0, 14). So it wasn't any work as far as evaluating an expression, and we weren't on the coordinate plane. We still needed to understand that relationship with-- when I have an X-intercept-- you must remember Y is 0. When you have your Y-intercept, you must remember that X is 0. Then once you're presented with a list of points, it's easy to locate which is which. Okay? Let's keep moving. This time we have a list of four points, but our process is still the same. They're asking us to find the X- and Y-intercepts. I know for my X-intercept-- I always start with this-- I know I need some numerical value for X, but Y is 0. For my Y-intercept, I know that I need-- I know that I need some-- X is 0, and I need some numerical value for Y.

(Describer) The ordered pairs are 0, negative 3; 6, 0; 2, negative 2; and 8, 1.

(female narrator) The ordered pairs are (0, negative 3),

(6, 0), (2, negative 2), and (8, 1). So X-intercept, I know Y's got to be 0, so there we go. It's my second point listed here. So X-intercept: (6, 0). My Y-intercept, I know that X must be 0 and I can look and see it's the first point listed here. So Y-intercept is at 0, negative 3. All right? Good job on that one. It's time for you to try one all on your own. Press pause, take a minute and try to locate the X- and Y-intercepts of this function. When you're ready to check your answer, press play.

(Describer) Title: Identify the x and y intercepts. The ordered pairs are 2, 9; 3, 0; 5, negative 18; and 0, 27.

(female announcer) Identify the X- and Y-intercepts.

The ordered pairs are: (2, 9), (3, 0),

(5, negative 18), and (0, 27). Identify the x- and y-intercepts. All right, let's see how you did. First, I'm writing my little shorthand note. X-intercept, X is some number, Y has to be 0. Y-intercept, X is 0, Y is some number. Now, don't think that that number can't be 0, okay? It can sometimes, but you must remember that for your Y-intercept, X must be 0, okay? The Y-intercept could also be 0, but the X has to be 0. Then the reverse of that for your X-intercepts, okay? For my X-intercept, I know I'm looking for the point where Y is 0, and it's here. So my X-intercept is (3, 0). My Y-intercept, I know I'm looking for X to be 0, and here we go.

It's that last point listed: (0, 27). All right. Good job on that one. We have reached the end of our lesson, guys. Great job solving problems involving finding X- and Y- intercepts of linear functions. You saw how all your prior knowledge of algebra and pre-algebra, the coordinate plane, and evaluating expressions all came together to help you through this one. See you soon for more Algebra 1. Bye!

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