Question

What is the equation, in standard form, of a parabola that contains the following points?
(-2,18),(0,2),(4,42)

Answer 1

so we know that the equation of a parabola is \(\bf y = ax^2+bx+c\)
so we would like to know what "a" "b" and "c" are

well.. we have 3 points.... so let's plug those guys in

\(y = ax^2+bx+c\qquad \qquad (-2,18)\quad (0,2)\quad (4,42)\\ \quad \\
\begin{array}{llll}
(-2,18)\\
18 = a(-2)^2+b(-2)+c\implies &\bf 4a-2b+c=18\\ \quad \\
(0,2)\\
2 = a(0)^2+b(0)+c\implies &\bf c = 2\\ \quad \\
(4,42)\\
42 = a(4)^2+b(4)+c\implies & \bf 16a+4b+c = 42
\end{array}\\ \quad \\
\textit{we know that }\quad c = 2\quad \textit{so let's use that in the 1st and 2nd equations}\\
4a-2b+(2)=18\implies \color{blue}{4a-2b=16}\\ \quad \\
16a+4b+(2) = 42\implies \color{blue}{16a+4b=40}\)

Answer 2

so as you can see, the 3 points given, give a system of equations of 3 variables, from our parabola standard form template

and you'd just have to solve the system of equations for the variables, in this case, since the 2nd equation gave us only c = 2, then we can substitute that in the 1st and 3rd equations and end up with only a system of 2 equations

and from there, you can substitute or eliminate, to get either "a" or "b" and then get the other variable

then once you know the values for "a" "b" and "c", just plug them in, in the standard form

Answer 3

Thank you! That helped me sooo much!!

Answer 4

yw