Answer
The equation of the parabola is $$y=2x^2-x$$
Work Step by Step
$$y=f(x)=ax^2+bx$$
- Derivative of $f(x)$: $$f'(x)=2ax+b$$
1) $f'(x)$ is the slope of the tangent line to the parabola at point $A(x, y)$.
So, since the tangent line to the parabola at point $(1,1)$ has the equation: $$y=3x-2$$
we find that $f'(1)=3$, or in other words, $$2a\times1+b=3$$ $$2a+b=3\hspace{1cm}(1)$$
2) There is a tangent line to the parabola at point $(1,1)$. That means point $(1,1)$ must lie in the parabola. Therefore, $$a\times1^2+b\times1=1$$ $$a+b=1\hspace{1cm}(2)$$
Subtract (2) from (1), we have $$(2a-a)+(b-b)=3-1$$ $$a=2$$
So, $$b=1-a=1-2=-1$$
Therefore, the equation of the parabola is $$y=2x^2-x$$
