Question

find a function y=ax^2+bx+c whose graph has an x intercept of 1, a y intercept of 2, and a tangent line with a slope of -1 at the y intercept

Answer 1

Let
\[
f(x)= a x^2 + b x +c\\
f(0)=2= c\\
f(x)= a x^2 + b x +2
\]
We still need to find a and b. Can you do it?

Answer 2

We have
\[
f(1)=0 = a+ b +2\\
a+ b =-2
\]
We need another relation between a and b.
We still have one property that we have not used.

Answer 3

\[
f'(x) = 2 a x + b\\
f'(0)= -1= b\\
b=-1\\
\]
How much is a?

Answer 4

a+b =-2
a-1=-2
a=-1
\[
y=f(x)= - x^2 - x +2
\]