# How to get a parabola from 2 points and a slope

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?

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I'm sorry... the slope where is 0.364? A slope cannot "start at $(0,0)$ and go[] through A" because slopes are not lines: slopes are numbers associated to lines. The slope of a graph at a point is the slope of the tangent at that point. –  Arturo Magidin Mar 12 '12 at 18:16
The "angle" of the slope, as in the function is f(x)= 0.364x + b –  Delusional Logic Mar 12 '12 at 18:17
Your description of the slope is incoherent; a slope is not a line. The function $f(x)=0.364x+b$ is not a slope, it's not an angle. One cannot speak about "the angle of the slope", because the slope is the rise of a line divided by its run; it's a number; it doesn't have an angle. –  Arturo Magidin Mar 12 '12 at 18:19
I'll just write what i have instead of what i calculated myself then. The "line" starts at (0,0), going through (8, 2.912) giving it a climb of 0.364y per x –  Delusional Logic Mar 12 '12 at 18:22
Fair enough; then perhaps it's just a translation error. But a single number cannot refer to a something that starts at $(0,0)$ and goes through $A$ in any case (either as "angle" or as "slope"). And a line cannot be given by a single number. –  Arturo Magidin Mar 12 '12 at 18:31

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 $b$.

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You assumption is correct! I am currently very confused about these 3 terms. (angle, line, slope) but i'll figure it out. –  Delusional Logic Mar 12 '12 at 18:46
@DelusionalLogic: One thing that may help: find the term in Danish Wikipedia and then look at the corresponding article in English WP. –  Brian M. Scott Mar 12 '12 at 18:49

Assuming the parabola is of the form $y = a(x-b)^2 + c$, the fact that $y(8) = y(16)$ tells you that the vertex of the parabola is halfway between $x=8$ and $x=16$; that is, at $x=12$. So $b=12$.

The slope of the parabola at the point $x=k$ is given by $2a(k-b)$ (this follows by taking the derivative: $y' = 2a(x-b)$, so evaluating at $x=k$ gives the quantity I just gave you). If you know $k$, then since we now know $b$ we can obtain the value of $a$. That will tells us the value of $a$ and $b$, and we only need to figure out the value of $c$. Since we know the value of $y$ at specific points, we can plug them into the formula and solve for $c$.

Once you know $a$, $b$, and $c$, you know the parabola.

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