I need to get the equation of a parabola, i have 2 points and a slope
The 2 points are as follows:
- A = (8, 2.912)
- B = (16, 2.912)
- Slope = 0.364
The slope starts at (0,0) and goes through A
Technically i have an additional slope, but that is just mirrored for B.
How do i get the parabola connecting these two points, where the slope is a tangent?
Best Answer
I’m assuming that when you say that the ‘slope’ starts at $(0,0)$ and passes through $A$, you mean that the tangent line to the parabola at $A$ goes through the origin. Since $\frac{2.912}8=0.364$, that line does in fact have a slope of $0.364$, but you should not confuse the line with its slope, any more than you would confuse yourself with your height or weight.
You know that the axis of the parabola lies midway between $A$ and $B$, so it’s the line $x=12$. This means that the equation of the parabola has the form $$y=a(x-12)^2+b\tag{1}$$ for some constants $a$ and $b$. You know that $(8,2.912)$ lies on the curve; substituting that into $(1)$ gives you one equation involving $a$ and $b$. Differentiating $(1)$ and substituting for the slope of the tangent at that point gives you a second, and you should then be able to solve the resulting system for $a$ and $b$.