In quadratic regression with one predictor, what is the general form of the regression equation relating response $y$ to predictor $x$?

• A. $y={\beta }_0+{\beta }_{1}x$
• B. $y={\beta }_0+{\beta }_1{x}^3$
• C. $y={\beta }_0+{\beta }_1{x}^2$
• D. $y={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2$

Answer: (D)

Verified step by step guidance

1. Understand that quadratic regression models the relationship between the response variable \(y\) and the predictor variable \(x\) by including both a linear term and a squared term of \(x\) to capture curvature.
2. Recall that the general form of a simple linear regression is \(y = \beta_0 + \beta_1 x\), where \(\beta_0\) is the intercept and \(\beta_1\) is the slope coefficient for \(x\).
3. In quadratic regression, we add a term involving \(x^2\) to the model to allow for a parabolic relationship. This means including a coefficient \(\beta_2\) multiplied by \(x^2\).
4. Write the general quadratic regression equation as \(y = \beta_0 + \beta_1 x + \beta_2 x^2\), where \(\beta_0\) is the intercept, \(\beta_1\) is the coefficient for the linear term, and \(\beta_2\) is the coefficient for the quadratic term.
5. This equation allows the model to fit data where the effect of \(x\) on \(y\) is not just a straight line but can curve upwards or downwards depending on the sign and magnitude of \(\beta_2\).