I have just started linear algebra (first week, we just finished going over gauss-jordan elimination) and am running into some difficulties in regards to a homework concept.
The stated problem:
"The curve $y = ax^2 + bc + c$ shown in the figure (I can't get this figure for you sorry!) passes through the points $(x_1, y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the coefficients $a, b,$ and $c$ are a solution of the system of linear equations whose augmented matrix is $ \left[\begin{array}{rrrr} x_1^2 & x_1 & 1 & y_1 \\ x_2^3 & x_2 & 1 & y_2 \\ x_3^2 & x_3 & 1 & y_3 \end{array}\right] $ "
I realize that the variables in this matrix are actually the a, b, c's and that the x's are in fact numbers, but I am really kind of lost on how to proceed with this type of problem. I am having a bit of a mental block on what is being shown to me here, and more so what I am aiming to do. We didn't cover anything like this in class and the next problem set has another similar problem in it, which I will briefly cover as well.
"Find the coefficients $a, b, c,$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation $y = ax^3 + bx^2 + cx + d$ " I am sorry I can't get the picture, if this is pivotal I can scan them in, there are 4 points given on the curve in the picture $(0,10), (1,7), (3,-11),$ and $(4,-14)$
Mostly I am wondering how to even approach these types of problem concepts and what I am supposed to be looking for? I hope I am making sense.