1. derivative application

Show that the tangent to P: y= ax^2+bx+c with gradient m has y intercept c-(m-b)^2/4a. thank you.

2. Originally Posted by slaypullingcat
Show that the tangent to P: y= ax^2+bx+c with gradient m has y intercept c-(m-b)^2/4a. thank you.
m = 2ax + b => x = ....

Substitute that expression for x into y= ax^2+bx+c to get an expression for y. Now you know a point (x1, y1) on the tangent.

Then the tangent is y - y1 = m(x - x1).

Now substitute x = 0 and solve for y.

3. Hello, slaypullingcat!

The procedure is simple, but there's a LOT of algebra!
I was lucky to get the answer on the first try . . .

Show that the tangent to: . $y \:=\:ax^2+bx+c$ with gradient $m$

. . has y-intercept: . $c-\frac{(m-b)^2}{4a}$

The derivative of: . $y \:=\:ax^2+bx+c$ [1] .is: . $y' \:=\:2ax + b$

If the gradient is $m\!:\;\;2ax + b \:=\:m \quad\Rightarrow\quad x \:=\:\frac{m-b}{2a}$

Substitute into [1]: . $y \:=\:a\left(\tfrac{m-b}{2a}\right)^2 + b\left(\tfrac{m-b}{2a}\right) + c \quad\Rightarrow\quad y \;=\;\frac{m^2-b^2 + 4ac}{4a}$

The tangent has point $P\left(\tfrac{m-b}{2a},\;\tfrac{m^2-b^2+4ac}{4a}\right)$ . and slope $m.$

Its equation is: . $y - \tfrac{m^2-b^2+4ac}{4a} \;=\;m\left(x - \tfrac{m-b}{2a}\right)$

. . $\text{which simplifies to: }\;y \;=\;mx + \underbrace{c - \tfrac{(m-b)^2}{4a}}_{y\text{-intercept}}$

4. That was a quality post, thank you very much. I understand it now.