Functions

[quick plank]

Let be the family of functions.Show that the parabolas associated with these functions pass through two fixed points.
b) To determine m for which the top of the parabola is located in quadrants 2 or 3.

[hexed aurora]

Is this $f_m(x) = (m-1)x^2 +(3m-1)x +(2m+1)$

where $m\in\mathbb{R}\setminus{1}$?

[gloomy heart]

Jay

[hexed aurora]

If all the graphs of $y=f_m(x)$ passes through

two fixed points $A$ and $B$, then $A$ and $B$ must

lie on every curve $y=f_m(x)$ for all $m\in\mathbb{R}\setminus{1}$.

Hence, you can choose any value of $m\in\mathbb{R}\setminus{1}$

and you will get a curve on which $A$ and $B$ lie.

So, choose any two values of $m\neq 1$ and solve

the resulting curves simultaneously for $A$ and $B$.

You can then prove that $A$ and $B$ lie on every

curve in this family by showing the coordinates

of $A$ and $B$ satisfy the general equation above

when treating $m$ as an unknown.

[gloomy heart]

Jay

[wanton linden]

this is a first approach but generally if A(a;b) is a fixed point then f_m(a)=b=constant, since m is a variable, then the equality holds for every m different than 1 if and only if the factor of m in f_m(x) cancels out

this is helpful since it's direct

meanwhile when taking two values of m, there's a possibilty (in general cases) that there are more than 1 sol when only one is right