Question 246783: Write a equation whos graph has a vertex of (-3,-2) and passes through the point (1,-10) Answer in vertex form Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! You don't mention what kind of equation you need. Since you have listed this problem under Quadratic Equations I'll assume that this is what you need. The graphs of these equations are parabolas. But you don't mention whether you want a vertically oriented parabola or a horizontally oriented parabola. I'll start with a vertically oriented and then repeat the solution for a horizontally oriented one.
A useful form (which you describe as "vertex form") for the equation of a vertically oriented parabola is:
Note: Different books may use different variations of this equation. Some will have the k on the left: . Some will have something other than 4p in front of . But regardless of the variation, the h and the k represent the x and y coordinates of the vertex: (h, k).
Since we want a vertex of (-3, -2) we want h = -3 and k = -2. Substituting these into the equation we get:
which simplifies to:
All we need now is a number for p. We can use our "other" point, (1, -10), for this. Since this other point needs to fit the equation (since it is supposed to be on the parabola), we should be able to substitute its coordinates into the equation:
The only unknown in this equation is p. We should be able to solve for p. Start by simplifying:
Add 2 to both sides:
Divide both sides by 64:
Now that we have p we can go back and write the full equation:
which simplifies to
or
If you need a horizontally oriented equation, the useful form is:
We use the same steps as above so I will not comment this very much.
Substitute in for the vertex:
Using p and the vertex to write the desired equation:
which simplifies to:
(