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Find the vertex, focus, and directrix of the parabola. Then sketch the parabola.$$y^{2}=3 x$$...

Question

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola.$$y^{2}=3 x$$

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$y^{2}=3 x$$



Answers

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$y^{2}=3 x$$

He has clears the one you married here. So we have two acts is equal to negative Wise square. And we get to rewrite this as negative two acts is equal to y square. Can we get Why squared to be equal to four times negative 1/2 times x our vertex I'll put us V. We have zero comma. Zero our focus. We get negative 1/2 comma zero in our direct tricks is access equal to 1/2. When we drop our graph. Look something like this. This apart is negative 0.5, comma zero and our line X is equal to one Have

So we have the following equation and we want Thio find all the pieces of the proble and schedule. So first we just settle for other X squared r Y squared to get it to look like one of the standard for mills. So since we have X squared in our equation, that's all for that. So first we subtract a part of the other side and then divide by the three so we get X squared is equal to negative 8/3. Why, Um, so since we have on X squared, we know it's gonna be either open up or open down since our coefficients negative. That tells me that our problem is gonna be opening toe onward. Ah, so now this looks like Formula One in the book. Um, so we're gonna compare it to X squared equals four p. Why? So that tells me that four p has to be equal to negative eight or three. So now I can solve for P by dividing both sides by four, which is the same is multiplying by 1/4. So this tells us that P is equal to minus two over. So, um, the equation for this proble here has Vertex zero since we're not subtracting anything, um, from the X and A Y, the focus is located at zero comma p, which in our case, is zero negative 2/3. And the direct tricks remember is a line. And for this equation that's located at why equals negative p um, which in this case for us is negative negative 2/3. So becomes a positive 2/3. So now, um, we're in good shape to draw sketch. So first we know over Texas as 00 the focus is at zero negative. 2/3 on the direct tricks is that why equals 2/3 So that's gonna be a horse on a line through 2/3 on the Y axis. And we know that the Vertex, um, are sorry. The problem always opens away from the direct tricks and contains the focus inside me. So this is a sketch of our craft

It's clear. So when you read here so we have the proble at square, it's equal to eat. Why we could get this from the equation. X plus two square is equal to eight times Y minus three. Her X square is equal to eight. Why? It could be rewritten. Aspect Square is equal to four times two. Why the Vertex here is zero comma zero the Fergus zero comma too. In the direct tricks is why is equal to negative too. Yeah. So if we shift this three up, we get X square is equal to eight times why minus three. So the bird tracks becomes zero comma three the Fergus zero comma five. And the direct tricks is why equals one. And if we shifted to to the left, we get the Graff X plus two square is equal to eight times Y minus three. So the vertex becomes negative to calm a three. Fergus is negative. Two comma five. In the direct tricks, it remains the same. No, we draw our graph. They look something like this. Our direct tricks. Why is equal to one Biograph? It's gonna be in black. This is negative. Two comma five

So we have a following equation for a parabola. And so first we want to try to write it, um, and standard form, so we won't try to get it to look like Why minus k squared equals four p times X minus h where h k is the center. We're sorry. The vortex. Ah. So first we need to get the coefficient of this X to be so we can write it as y minus two squared is equal to two times X plus. And then if we factor out to to get us back to one, we need to multiply by 1/2. So we get explosive half. So now this looks like, um, the form we want. So in this case, so what does age and what is K? Um, we have to take the opposite of the, uh, number that I'm underlining, so h instead of positive. 1/2 is negative 1/2 and kay is positive, too. Since it's the number that we're subtracting. That's why we flip the sign. So this is our vortex s. So now we need to find P to find our directorates and problem, So we need to look at the coefficient on the outside, um, of the ex term, which is two. And we need to set it equal to four people. So if we do that Ah, well, you can divide both sides by four too easily. See that P is 1/2. Um, so our focus is H plus P comma cake using the formula in the book lips. So in this case, age plus P is negative. 1/2 plus 1/2 which is zero on K is too. I'm on the direct tricks is that ah, located at X equals H minus P, which in our case is negative 1/2 minus one, Huh. Which is negative one. So let's go ahead and try to sketch a graph. So first, we could draw in the Vertex at negative 1/2 common, too. So we have our point here, and then we can plot our focus at 02 which is here and then, Ah, since the coefficient four p is positive, we know a problem opens to the right. So it's going to look something like this, and our direct tricks is ah h equals negative one. Our X equals negative one. So it's a vertical line going through negative one on the X axis


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