Question

a parabola contains the points (-1,0), (0,-3), and (1,-2). What is the equation of this parabola in standard form

Answer 1

\(\bf y = ax^2+bx+c\qquad \qquad (-1,0)\qquad (0,-3)\qquad (1,-2)\\ \quad \\
\textit{let us use the form or template above for each point}\\ \quad \\


\begin{array}{llll}
(-1,0)\qquad x = -1\qquad y = 0\\ \quad \\
0=a(-1)^2+b(-1)+c\implies &\bf \color{blue}{a-b+c =0}\\ \quad \\
(0,-3)\qquad x = 0\qquad y = -3\\ \quad \\
-3 = a(0)^2+b(0)+c\implies &\bf \color{blue}{c = -3}\\ \quad \\
(1,-2)\qquad x = 1\qquad y = -2\\ \quad \\
-2 = a(1)^2+b(1)+c\implies &\bf \color{blue}{a+b+c=-2}
\end{array}\)


so you see, you have a system of equations with 3 variables

notice that the 2nd one gives us c = -3

so we can use that in the 1st and 3rd one

Answer 2

\(\bf a-b+c=0\implies a-b+(-2)=0\implies a-b=2\\
a+b+c=-2\implies a+b+(-2)=-2\implies a+b=0\)

and now you're left with a system of equations of 2 variables, so you can solve that by substitution or elimination, to get "a" and "b", and then use that to get "c" by plugging them in either of the 1st or 3rd equations

Answer 3

The three points:
(-1,0) (0,-3) (1,-2)
we use the general formula for a parabola
y=ax2+bx+c
and plug in the values for each of the 3 points, producing three equations

A) 0 = a(-1²) +b(-1) + c

B) -3 = a*0 + b*0 +c

C) -2 = a(1²) + b*1 +c

From equation C we know that c=-3
we rewrite the equations
A) 0 = 1a -b -3
B) -3 = -3
C) -2 = a + b -3

From equation A we see that
a = b+3
putting this into equation C
-2 = b+3 + b -3
-2 = 2b
b=-1
Using Equation A
A) 0 = 1a --1 -3
0 = 1a +1 -3
2 = 1a
a = 2

So equation equals
-2 = 2x² -x -3
SO equations equals

2x² -x -1

(I'd better check that answer).

Answer 4

Thank you so very much for your help. :)