How can I write a parabola in standard form?
Standard Form: Y = Ax^2 + Bx + C.
To write a parabola in standard form, you would use the equation:
y = ax^2 + bx + c
where 'a', 'b', and 'c' are coefficients that determine the shape, position, and orientation of the parabola.
Here are the steps to write a parabola in standard form:
1. Identify the vertex of the parabola. The vertex is the point on the parabola where it reaches its maximum or minimum value. It is denoted as (h, k).
2. Use the vertex form of the parabola equation, which is given by:
y - k = a(x - h)^2
In this form, (h, k) represents the vertex of the parabola.
3. Expand the equation and simplify it to obtain the standard form. Expand the equation (x - h)^2 using the FOIL method:
(x - h)(x - h) = x^2 -2hx + h^2
Then, multiply through by 'a':
a(x^2 - 2hx + h^2) = ax^2 - 2ahx + ah^2
Finally, rearrange the terms to obtain the standard form:
ax^2 - 2ahx + ah^2 - y + k = 0
4. Simplify the equation further if possible. Combine like terms and adjust the signs if needed to make the equation match the standard form:
ax^2 + (-2ah)x + (ah^2 - y + k) = 0
This is the standard form of a parabola equation.
Remember, 'a' determines the shape of the parabola (a positive value opens upwards, and a negative value opens downwards), while 'h' and 'k' determine the position of the vertex.