SM2 14.2-3 Finding the Information of a Horizontal Parabola

追加: 2026-05-09

[00:00:00]Hello and welcome. So it says given the equation  of a parabola below find the following. So first thing we've got to find before we do any  of this stuff, find which direction your parabola is going. Right, you have four options  it's either going up, down, left or right, one of the four. Now how you find out which one  is which is you got to look at your equation.

[00:00:24]All right, we have two kinds of equations. Right,  you have x minus h squared equals 4p, k minus-- sorry, not k, y minus k, there we go, or you have  y minus k squared equals 4p, x minus h. Now the only difference between these is that as you  look from one equation to the other you notice one of them is going to be squared, and the one  that isn't squared has the 4p attached to it.

[00:00:54]Okay, now if x is squared that is a vertical  parabola. If y is squared then it's horizontal.

[00:01:04]Okay, so if x is being squared then  it's either going up or it's going down, one or the other not both. One or the other.  And then if y is being squared which we notice is what we do have, then your option is  either it's going left or it's going right.

[00:01:20]And how we determine it's going left or right is  you look at the 4p value, and then that tells you whether you're going left or right. So x is left  or right. Okay, so if you have like zero in the middle left is negative, right is positive. Well  if you have a negative number, negative numbers go to the left and so your parabola the one we're  graphing, [cough] excuse me, is also going to the left.

[00:01:46]Okay, so we have a parabola and it's going to the  left we know that for sure because our y is being squared and our 4p value is negative. Okay, that  is how we know. Okay, if the 4p value is positive you'd be going to the right, if x was being  squared. Right, if you had this or that. Okay, or if your 4p value is positive positive is  going up right for your y values you would be going up on a parabola. Okay, if your 4p value  is negative you're going down with the parabolas.

[00:02:18]So that's how you could tell your direction  difference for the other versions as well.

[00:02:22]All right, so for this one vertex is always  going to be your h comma k. All right, which means that now all the other ones are kind  of being determined similarly as well. h and k are just the numbers being shifted right the numbers  being added subtracting to your x and y. Right, but notice it's subtraction in your equation  meaning you got to flip the signs of what you see. So if you have a plus one that means  you're actually going to be minusing one as you're-- for part of your y values. Right, if  your x has plus six you're going to be doing minus six. Okay, if you add x minus 6 then it'd be plus  six. Right, whatever the sign is do the opposite.

[00:03:02]Okay, now your focus you may remember is that  point inside the direction of opening. Now what's cool is that it's just a p distance away  from the vertex. Right, so if that's your focus here's your vertex. Okay, and we know that because  it's horizontal what's going to be happening is that we're going to be adding our h value and add  it to whatever we find our p-value to be. And then we're not moving up or down, if you're just moving  to the left that's not going up or down, k value stays the same. So here for p you're going to see  that right in front of our x minus h parentheses, so your 4p value is negative 68 here. So that  means 4p equals negative 68. If we want to solve for our p-value that's dividing by 4, both sides,  and then we get that p is negative 17. Okay, which means that we go negative 17 to the left from  the vertex. So that means that we have starting at negative 6, we're going negative 17 more to the  left but our y's stay the same. Well negative 6 minus 17 that's negative 23. So negative 23 comma  negative 1 is the coordinate point for your vertex (Mistake: meant focus).

[00:04:17]Directrix is behind your parabola so in this  case it's going to be to the right of it, and that's also a p value away from your vertex.  Now think about it this way, if you have like a x-axis right you'll notice your directrix would  be running through the x-axis that means that your equation is going to be x equals for  the line. And then how we find that is that we know that it's just going to be a, for this  one, it's going to be x equals our h value of our vertex, right, x value of the vertex and we're  essentially going to be going the other direction from-- as our focus did. So we're going to subtract  p instead. So that means that in this case we have negative 6 minus a negative 17 or in other words  we're going to be adding 17. So that means that negative 6 plus 17 that is 11. There is your  directrix. Okay, now this is a little bit of a vocab review. Axis of symmetry is like if you were  to fold your parabola in half where would that, like the fold line be that is your axis of  symmetry. So you'll notice that it's going the other direction so instead of x equals it's going  to be y equals because it'll cross the y-axis if you were to keep going with it. Okay, and then  from there you'll notice that it's matched it's like hitting through the focus and through your  vertex so it'll just match whatever the y values of those ones are, in other words, negative one.  Bam. Okay, and then from there you have a focal diameter. Your focal diameter that's just a  fancy word for 4p. Now keep in mind though that a diameter is a length and lengths cannot  be negative so your focal diameter well we see that in the 4p value it can't be a negative number  because it's a length so you just put positive 68 instead. And that would be all of the information  for this thing. All right, thanks for watching.