Find the Function Rule table[[x,y],[0,0],[1,-2],[2,0]]

Simplify each equation.

Rewrite the equation as ${\displaystyle b=0}$.

Replace all occurrences of ${\displaystyle b}$ with ${\displaystyle 0}$ in each equation.

Simplify.

Rewrite the equation as ${\displaystyle a=-2}$.

Replace all occurrences of ${\displaystyle a}$ in ${\displaystyle 0=2a}$ with ${\displaystyle -2}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle -2}$.

Since ${\displaystyle 0\ne -4}$, there are no solutions.

No solution

Calculate the value of ${\displaystyle y}$ when ${\displaystyle a=-2}$, ${\displaystyle b=0}$, and ${\displaystyle x=0}$.

If the table has a linear function rule, ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ value, ${\displaystyle x=0}$. This check passes since ${\displaystyle y=0}$ and ${\displaystyle y=0}$.

Calculate the value of ${\displaystyle y}$ when ${\displaystyle a=-2}$, ${\displaystyle b=0}$, and ${\displaystyle x=1}$.

If the table has a linear function rule, ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ value, ${\displaystyle x=1}$. This check passes since ${\displaystyle y=-2}$ and ${\displaystyle y=-2}$.

Calculate the value of ${\displaystyle y}$ when ${\displaystyle a=-2}$, ${\displaystyle b=0}$, and ${\displaystyle x=2}$.

If the table has a linear function rule, ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ value, ${\displaystyle x=2}$. This check does not pass, since ${\displaystyle y=-4}$ and ${\displaystyle y=0}$. The function rule can’t be linear.

Since ${\displaystyle y\ne y}$ for the corresponding ${\displaystyle x}$ values, the function is not linear.

The function is not linear

Simplify each equation.

Rewrite the equation as ${\displaystyle c=0}$.

Replace all occurrences of ${\displaystyle c}$ with ${\displaystyle 0}$ in each equation.

Simplify.

Solve for ${\displaystyle a}$ in the second equation.

Replace all occurrences of ${\displaystyle a}$ in ${\displaystyle 0=4a+2b}$ with ${\displaystyle -2-b}$.

Simplify ${\displaystyle 4(-2-b)+2b}$.

Solve for ${\displaystyle b}$ in the third equation.

Replace all occurrences of ${\displaystyle b}$ in ${\displaystyle a=-2-b}$ with ${\displaystyle -4}$.

Simplify ${\displaystyle -2-(-4)}$.

Calculate the value of ${\displaystyle y}$ such that ${\displaystyle y=a{x}^2+b}$ when ${\displaystyle a=2}$, ${\displaystyle b=-4}$, ${\displaystyle c=0}$, and ${\displaystyle x=0}$.

If the table has a quadratic function rule, ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ value, ${\displaystyle x=0}$. This check passes since ${\displaystyle y=0}$ and ${\displaystyle y=0}$.

Calculate the value of ${\displaystyle y}$ such that ${\displaystyle y=a{x}^2+b}$ when ${\displaystyle a=2}$, ${\displaystyle b=-4}$, ${\displaystyle c=0}$, and ${\displaystyle x=1}$.

If the table has a quadratic function rule, ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ value, ${\displaystyle x=1}$. This check passes since ${\displaystyle y=-2}$ and ${\displaystyle y=-2}$.

Calculate the value of ${\displaystyle y}$ such that ${\displaystyle y=a{x}^2+b}$ when ${\displaystyle a=2}$, ${\displaystyle b=-4}$, ${\displaystyle c=0}$, and ${\displaystyle x=2}$.

If the table has a quadratic function rule, ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ value, ${\displaystyle x=2}$. This check passes since ${\displaystyle y=0}$ and ${\displaystyle y=0}$.

Since ${\displaystyle y=y}$ for the corresponding ${\displaystyle x}$ values, the function is quadratic.

The function is quadratic