Hey everyone, and welcome back. So let's give this example a try. Here we're asked to identify the vertex, focus, and directrix for each of these parabolas below, and we'll start with parabola a. I notice about this parabola is that we have a y², and because we have a y² that means we're dealing with a horizontal parabola. The equation for a horizontal parabola is 4p⁢⁡⁢⁡⁢⁡⁢⁡x−h⁡=y−k2. So what we can do is use this equation as a reference for each of these examples. Looking at this first example, I notice nothing is subtracted from the x, and nothing is subtracted from the y. So in this case, our h would be 0 and our k would be 0. The vertex of this parabola is going to be at the origin (0,0). The vertex is pretty straightforward for this parabola. Now, we need to also find the focus, and to find the focus, we can use this p value. So what I can do is set everything that's in front of the x, the x minus h here, equal to everything in front of the x there. So, that means that 4p⁢⁡⁢⁡⁢⁡=4, and what I can do is divide 4 on both sides to get p by itself. Now we're going to get that p is 4 divided by 4, which is 1. Now notice how our p value came out to positive 1. Since we have a positive p value, that means that our horizontal parabola is going to open to the right. Since we're opening to the right, that means our focus is going to be somewhere in there, and our directrix is going to be somewhere back here, just for example. So, what we can do is take a look here and see how things are going to behave based on this p value. Now the absolute value of positive one is also just 1, and so looking at our vertex, if we were to go 1 unit to the right, our focus would be at the position 10, because horizontally we'd be at a position of 1 for our focus, and our directrix line which is going to be x is equal to the line would be one position back from 0 which is at negative one. So, this would give us the vertex, the focus, and the directrix for this first parabola. But now, let's take a look at our next parabola, parabola b. In this equation, I notice that some number is being subtracted from the y. Because of this that means we are not going to be dealing with a parabola at the origin. So, what I can do is take a look at what we have in front of x, and I see that nothing is being subtracted from the x. So, that means h is 0. And I can see that 2 is being subtracted from the y, so k is 2. Our vertex is going to be at the horizontal position of 0 and the vertical position of 2, and that's our vertex. Now, from here I need to find the focus point, and to find the focus point I need to first find the &#p value. So I'll set 4p⁢⁡⁢⁡⁢⁡ in front of everything that's in front of the x, which is going to be 9. If I divide 4 on both sides that'll get the force to cancel, giving me that &#p is equal to 9 over 4. Now notice that once again we got a positive &#p value, and because we got a positive &#p value, our parabola is going to open to the right. And that means our focus will be somewhere there, for example, and our directrix is going to be behind the graph. So, what we can do here is use this &#p value and go to the right to find our focus, and to the left to find our directrix. So our focus is going to be to the right horizontally we're at 0, and going to the right we'd be at 9 4ths by this &#p value 2, and by the way, we're doing this by the absolute value of &#&#&#p. So the absolute value of 9 4ths is just 9 4ths, but I need to go to the right by that many units, so that's why we ended up with this for our focus, and our directrix is going to be at x equals negative 9 fourth since we need to go to the left by 9 fourths which we're starting from 0. So that would be our directrix. So this would be the vertex focus and the directrix for parabola b. But now let's take a look at parabola c. Well, I can see once again that we have numbers that are being subtracted from the x and y, so that means we're not at the origin. Now looking at this equation, I can see that h corresponds with this 4. I can see that k corresponds with 2, but notice that we have a plus 2, and the equation says that we have y minus k. So because we have an opposite sign here, y plus 2² is the same thing as y minus negative two squared. So that means that our k is actually negative 2, so we need to put a negative number in 4 k. So our vertex is going to be at the position 4 negative 2. Now from here, I can find the focus and directrix and to find the focus I first need to find the &#p value. So we'll set 4p⁢⁡⁢⁡⁢⁡ equal to everything in front of the x which is 16 and If I go ahead and divide 4 on both sides that'll get the 4 to cancel giving me that &#p is 16 over 4 which is 4. Now notice our &#&#&#p value came out positive So that means that our parabola is going to open to the right since it's a horizontal parabola. Now to find the focus based on where our vertex is at, we're going to the right 4 units. And horizontally, we're at a position of 4. So going to the right 4 more units would put us at a position of 8. So we're going to be at 8 negative 2 for the focus. And as for the directrix, we need to go back 4 units, and if you take 4 and subtract out 4 we're going to get an x value of 0. Now, keep in mind that this is all again assuming the absolute value of &#&#&#&#&#&#&#&#&#&#&#&#&#&#p which is 4, but since I saw it was positive that's just gonna be the same number. So this is how you can find the vertex, the focus, and the directrix for parabola c. But now let's move on to our last parabola which is parabola d. What I can do is I can look at this equation and I can first try to find the vertex. Now our 'h' is going to be whatever is being subtracted from the x, which in this case is 1. And our k is going to be whatever subtracted from the y, but since I see nothing subtracted from the y, our k is 0. So that means that the vertex is going to be at 10. Now before I go any farther, what I'm actually going to do is draw a graph of what this parabola is going to look like. Because I think that and and we don't have to draw a graph for this, but I think that by looking at a graph, it's going to actually help us to really visualize where the focus and directrix end up. So I can see that our vertex is at the position 10, which would be let's just say right here on our graph. Let's say that's an x value of 1. So, this is where the vertex is going to be, and our parabola is either going to open to the right or to the left. And to figure that out, we need to find this &#p value, which will also tell us the distance to the focus and directrix. So to find &#p, well we've been doing this for that previous 3 problems, we just set 4p⁢⁡⁢⁡⁢⁡ equal to everything in front of x. So in this case our 4p⁢⁡ is going to be equal to negative 2, and I can go ahead and divide 4 on both sides of this equation that'll get the force to cancel there, giving me that &#&#&#p is equal to negative 2 over 4, which reduces to negative one half. But notice how the &#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#p value came out negative this time. And because the &#p value came out negative, that means our parabola is actually going to open to the left in this example. So we have a parabola that opens to the left. And because this parabola opens to the left, the focus is going to be somewhere to the left this time, and our directrix is going to be somewhere to the right. So it's going to be opposite of the other examples that we've had because now it opens the opposite direction. Let's see if we can find the focus point and directrix line. To find the focus point, well, I need to go to the left by a &#p amount of units. And notice that we start at a position of 1, and our focus point is going to be 1 half units to the left, but I need to find the absolute value of &#p first and the absolute value of negative one half is positive one half. So if I go to the left, one half units, well, one minus one half is one half. So that's going to be one half and 0 giving us our focus point onetwo comma 0. And to find our directrix line, I need to go to the right onetwo units. And to the right, well, we would have onetwo plus our position here which is 1. So we'd have 1 plus 1 half on the x axis which would put us at 3 half. So our directrix line is going to be x equals 3 halves, And that is how you can find the vertex, focus, and directrix for parabola d. So this is how you can solve these types of problems with horizontal parab...