Now, there are two forms of the parabola that we will be looking at. So if we just apply this-- but, you know, this is just kind of mindless application of a formula. One description of a parabola involves a point the focus and a line the directrix. You're gonna get an equation for a parabola that you might recognize, and it's gonna be in terms of a general focus, a,band a gerneral directrix, y equals k, so let's do that.
And the more positive that this number becomes when you multiply it by a negative, the more negative this entire expression becomes.
What I want to do in this video, it's gonna get a little bit of hairy algebra, but given that definition, I want to see, and given that definition, and given a focus at the point x equals a, y equals b, and a line, a directrix, at y equals k, to figure out what is the equation of that parabola actually going to be, and it's going to be based on a's, b's, and k's, so let's do that.
We MUST add first and then subtract. Conversely, light that originates from a point source at the focus is reflected into a parallel " collimated " beam, leaving the parabola parallel to the axis of symmetry.
Now, what I want to do is express the stuff in the parentheses as a sum of a perfect square and then some number over here.
The way I've just drawn it, yes, y is greater than k, so this is going to give us a positive value, and you need a non-negative value if you're talking about distances, but you can definitely have a parabola where the y-coordinate of the focus is lower than the y-coordinate of the directrix, in which case this would be negative.
That'll give me a positive 4 right there.
Now, in order to figure out the vertex, there's a quick and dirty formula, but I'm not going to do the formula here because the formula really tells you nothing about how you got it.
I woutould like to know how to find the equation of a quadratic function from its graph, including when it does not cut the x-axis.
In mathematicsa parabola is a plane curve which is mirror-symmetrical and is approximately U- shaped. A real-life example of a parabola is the path traced by an object in projectile motion. But if you just mindlessly apply this, you'll get-- what's b here?
So I can rewrite this as y is equal to-- well, I can factor out a negative 2. Part of a parabola bluewith various features other colours. The second form is the more common form and will require slightly and only slightly more work to sketch the graph of the parabola.
The focus does not lie on the directrix. And then you have this negative 2 out front multiplying everything, and then you have a negative 4 minus negative 4, minus 8, just like that.
Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side.
But what it does allow me to do is express this part right here as a perfect square. When x is equal to 0, y is equal to 8.
Due to the nature of the mathematics on this site it is best views in landscape mode. Here is the vertex for a parabola in the general form. So when x is equal to 0, we have 1, 2, 3, oh, well, these are 2. I added 4 and then I subtracted 4.Show transcribed image text Write the equation of the Parabola: with the vertex, (2, 5) and the y-intercept (0, 9).
It passes through 3 points (1, 4); (2, 9) and (5, 0). Parabola Calculator Deutsche Version This calculator will find either the equation of the parabola from the given parameters or the axis of symmetry, focus, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the entered parabola.
Video: Parabolas in Standard, Intercept, and Vertex Form By rearranging a quadratic equation, you can end up with an infinite number of ways to express the same thing. Given the information from the graph, we can determine the quadratic equation using the points of the vertex, (-1,4), and the point on the parabola, (-3,12).
Step 2: Sub Points Into Vertex. EXAMPLE 1 Finding the vertex, focus, and directrix, given an equation Find the vertex, focus, and directrix for the parabola y x2. Solution Compare y x2 to the general formulay a(x h)2 k.
We see thath 0, k 0, anda 1. So the vertex is (0, 0). Finding the Vertex and yIntercept from Factored Form 6 March 19, Finding the Vertex of a Parabola given a Factored Form Equation.Download