Question

Find a quadratic function to model the values in the table.x | y-1 -80 -23 -81. y =-2x^2 + 4x - 22. y=x^2+2x-23. y=-2x^2-4x-24. y=-x^2-2x+2

Answer 1

Step 1

When x=0, y=-2

`\text{The general equation of a quadratic function is y=ax}^2+bx+c`

Substitute for x=0,y=-2

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

Step 2

When x=-1,y=-8

`\begin{gathered} -8=a(-1)^2+b(-1)+c \\ -8=a-b+c \\ -8=a-b-2 \\ a-b=-8+2 \\ a-b=-6--(1) \end{gathered}`

When x=3,y=-8

`\begin{gathered} -8=a(3)^2+b(3)+c \\ -8=9a+3b-2 \\ 9a+3b=-8+2 \\ 9a+3b=-6---(2) \end{gathered}`

Step 3

From 1;

`\begin{gathered} a=-6+b \\ \text{Substituting for a in equation gives; 9(-6+b)+3b=-6} \\ -54+9b+3b=-6 \\ 12b=54-6 \\ 12b=48 \\ (12b)/(12)=(48)/(12) \\ b=4 \end{gathered}`

From equation 1;

`\begin{gathered} a-b=-6 \\ a-4=-6 \\ a=-6+4 \\ a=-2 \end{gathered}`

Therefore, the equation will be;

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