another tangent line: eqn of parabola through (0, 1) and tan

[orangecrush]

The question:

Find an equation of the parabola y = ax^2 + bx + c that passes through the point (0, 1) and is tangent to the line y = x - 1 at the point (1, 0).

Where do I start?

 

[galactus]

Since it passes through (0,1), set x=0 and we find that c=1.

\(\displaystyle a(0)^{2}+b(0)+c=1\), See?.

Find the derivative:

\(\displaystyle y'=2ax+b\)

The slope of y=x-1 is 1

Therefore, at (1,0) the slope of the parabola must be 1.

Set the derivative to 1, using x=1:

\(\displaystyle 2a+b=1\)

\(\displaystyle a(1)^{2}+b(1)+1=0\)

You have a small system to solve:

2a+b=1
a+b=-1

\(\displaystyle \L\\\begin{bmatrix}2&1&1\\1&1&-1\end{bmatrix}\)

After you find a and b, graph the line and the parabola.

 

[orangecrush]

Thank you