When testing the utility of the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$, the most important tests involve the null hypotheses $H _ { 0 } : \beta _ { 0 } = 0$ and $H _ { 0 } : \beta _ { 1 } = 0$.

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The most important test for the utility of a quadratic model involves testing the null hypothesis $H_0: \beta_2 = 0$, as this tests the significance of the quadratic term, which defines the model as quadratic.

When Testing the Utility of the Quadratic Model E(y)=β0+β1x+β2x2E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }E(y)=β0​+β1​x+β2​x2

When Testing the Utility of the Quadratic Model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$