I am given a graph of a parabola and I have to write a function in at least two forms that will produce the graph. I think that I can write the factored form first since I have the two x-intercepts but I don't know what the "a" is in the equation y= a(x-r1)(x-r2). How do I find the "a" so I can write the equation?
To find the value of "a" in the factored form of a parabola, you need to know one additional point on the graph. This could be either the vertex of the parabola or the coordinates of any other point on the curve.
Here's how you can find the value of "a" using the factored form and the vertex of the parabola:
1. Identify the vertex of the parabola from the graph. The vertex is the lowest or highest point on the curve, depending on whether the parabola opens upward or downward.
2. Once you have the vertex coordinates (h, k), substitute these values into the factored form equation: y = a(x - r1)(x - r2).
3. Use one of the x-intercepts (r1 or r2) along with the vertex coordinates to solve for "a". Substituting the x-intercept value for "x" in the equation and the corresponding y-coordinate of the vertex for "y", you get:
k = a(r1 - h)(r2 - h).
4. Solve the equation for "a" by isolating it. Divide both sides of the equation by (r1 - h)(r2 - h), then simplify to solve for "a":
a = k / (r1 - h)(r2 - h).
Once you find the value of "a", substitute it back into the factored form equation to get the complete equation of the parabola.