what is the equation, in standard form, of a parabola that contains the following points?
(-2-20). (0-4), (4-20)
If your bizarre notation indicates the points (-2,-20),(0,-4),(4,-20), then
4a-2b+c = -20
c = -4
16a+4b+c = -20
(a,b,c) = (-2,4,-4)
y = -2x^2 + 4x - 4
To find the equation of a parabola in standard form that passes through three given points, we can follow these steps:
Step 1: Write the equation in vertex form
Step 2: Convert the equation from vertex form to standard form
Let's go through these steps in detail:
Step 1: Write the equation in vertex form
The vertex form of a parabola equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
To determine the vertex (h, k), we can select any of the three given points. Let's use the point (0, -4):
Using the point (0, -4), we substitute the values of x and y into the equation:
-4 = a(0 - h)^2 + k
Step 2: Convert the equation from vertex form to standard form
The standard form of a parabola equation is given by:
ax^2 + bx + c = y
To convert the equation from vertex form to standard form, we need to expand the vertex form equation and simplify it.
Expanding the equation gives:
-4 = a(0 - h)^2 + k
-4 = a(h^2) + k
Now, let's substitute the coordinates of the other two given points (-2, -20) and (4, -20) into the expanded equation to form a system of equations.
For the point (-2, -20):
-20 = a(-2 - h)^2 + k
For the point (4, -20):
-20 = a(4 - h)^2 + k
These two equations can be solved simultaneously to find the values of a, h, and k.
Once the values of a, h, and k are known, we can substitute them into the vertex form equation and convert it to the standard form by expanding and simplifying the equation further.
By following these steps, you can obtain the equation in standard form for the parabola that passes through the given points (-2, -20), (0, -4), and (4, -20).