Question

Find an equation in standard form of the parabola passing through the points below. (1,8), (2,3), (4, - 31) The equation of the parabola is y=

Answer 1

`\begin{gathered} \text{ The standard equation of a parabola is} \\ y=ax^2+bx+c,^{} \\ \text{we have to find the values of a,b,c, for that, we will replace the given points } \\ \text{ in the equation, that results in} \end{gathered}`
`\begin{gathered} 8=a+b+c \\ 3=4a+2b+c \\ -31=16a+4b+c \end{gathered}`
`\begin{gathered} \text{ We will find the value, in the firs equation we do} \\ c=8-a-b \\ \text{then we replace in the second and third equation to get} \\ 3=4a+2b+(8-a-b)=8+3a+b \\ -31=16a+4b+(8-a-b)=8+15a+3b \\ \text{ then the resulting system is} \\ c=8-a-b \\ -5=3a+b \\ -39=15a+3b \end{gathered}`
`\begin{gathered} \text{Now, in the second equation we do} \\ b=-5-3a \\ \text{ And replace b in the third equation} \\ -39=15a+3(-5-3a) \\ -39=15a-15-9a \\ -39+15=15a-9a \\ -24=6a \\ a=-(24)/(6) \\ a=-4 \end{gathered}`
`\begin{gathered} \text{ now we replace a=-4 in } \\ b=-5-3a \\ b=-5-3(-4) \\ b=-5+12 \\ b=7 \end{gathered}`
`\begin{gathered} \text{and } \\ c=8-a-b \\ c=8-(-4)-7 \\ c=8+4-7 \\ c=12-7 \\ c=5 \end{gathered}`
`\begin{gathered} \text{and the equation of the parabola is } \\ y=ax^2+bx+c \\ y=-4x^2+7x+5 \end{gathered}`