Question

Find the equation of the quadratic function whose graph is a parabola containing the points (0,4), (2,10), and (−1,10).

Answer 1

A quadratic function is of the form

`\begin{gathered} ax^2+bx+\text{c} \\ \text{where} \\ a,b,\text{and c are real numbers} \end{gathered}`

let's substitute the values,

(0,4)​, (2,10)​, and (−1,10)

`\begin{gathered} 4=a(0)^2+b(0)+c \\ 4=c \end{gathered}`
`\begin{gathered} 10=a(2)^2+b(2)+c \\ 10=4a+2b+c \end{gathered}`
`\begin{gathered} 10=a(-1)^2+b(-1)+c \\ 10=a-b+c \end{gathered}`

Therefore, let's substitute the value of c in the other equations formed

`\begin{gathered} 10=4a+2b+4 \\ 6=4a+2b\ldots\ldots(1) \end{gathered}`
`\begin{gathered} 10=a-b+4 \\ 6=a-b\ldots\ldots\ldots(ii) \end{gathered}`

Let's combine the equation and find a and b

`\begin{gathered} 4a+2b=6 \\ a-b=6 \\ a=6+b \\ 4(6+b)+2b=6 \\ 24+4b+2b=6 \\ 24+6b=6 \\ 6b=6-24 \\ 6b=-18 \\ b=(-18)/(6) \\ b=-3 \end{gathered}`
`\begin{gathered} a-(-3)=6 \\ a+3=6 \\ a=6-3 \\ a=3 \end{gathered}`

Finally,

`y=3x^2-3x+4`