Question

select the correct answer from each drop-down menu what is the equation of the quadratic function represented by this table

Answer 1

Given data:

The given table.

The expression for the quadratic equation is,

`y=a(x-b)^2+c`

Substitute 0 for x and 3 for y in the above expression.

`\begin{gathered} 3=a(0-b)^2+c \\ 3=ab^2+c \\ c=3-ab^2 \end{gathered}`

Substitute 1 for x and 1.75 for y in the above expression.

`\begin{gathered} 1.75=a(1-b)^2+c \\ 1.75=a(1+b^2-2b)+c \end{gathered}`

Substitute (3-ab^2) for c in the above expression.

`\begin{gathered} 1.75=a(1+b^2-2b)+3-ab^2 \\ 1.75=a-2ab+3 \\ -1.25=a-2ab \\ 1.25=a(2b-1) \\ a=(1.25)/(2b-1) \end{gathered}`

Substitute -1 for x and 3.75 for y in the given quadratic equation.

`\begin{gathered} 3.75=a(-1-b)^2+c \\ 3.75=a(1+b^2+2b)+c \end{gathered}`

Substitute (3-ab^2) for c in the above expression.

`\begin{gathered} 3.75=a+ab^2+2ab+3-ab^2 \\ 3.75=a+2ab+3 \\ 0.75=a(1+2b) \end{gathered}`

Substitute 1.25/(2b-1) for a in the above expression.

`\begin{gathered} 0.75=(1.25)/(2b-1)(1+2b) \\ 1.5b-0.75=1.25+2.5b \\ -b=2 \\ b=-2 \end{gathered}`

The value of a is.

`\begin{gathered} a=(1.25)/(2(-2)-1) \\ =-0.25 \end{gathered}`

The value of c is,

`\begin{gathered} c=3-(-0.25)(-2)^2 \\ =3+1 \\ =4 \end{gathered}`

Thus, the given quadratic equationn is -0.25(x-(-2))^2 +4