Review Polynomials

To better understand the interpretation of polynomials, let’s see three plausible examples:

1. Positive linear, Negative quadratic

## 
## Call:
## lm(formula = health ~ exercise + I(exercise^2), data = d1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.92735 -0.59522  0.00662  0.74671  2.22454 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.13438    0.29274   3.875 0.000194 ***
## exercise       1.84092    0.17757  10.367  < 2e-16 ***
## I(exercise^2) -0.17177    0.02182  -7.873 4.98e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.02 on 97 degrees of freedom
## Multiple R-squared:  0.6524, Adjusted R-squared:  0.6452 
## F-statistic: 91.02 on 2 and 97 DF,  p-value: < 2.2e-16

2. Negative linear, positive quadratic

## 
## Call:
## lm(formula = fail_exam ~ anxiety + I(anxiety^2), data = d2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.9224  -3.8807   0.0049   3.6602  11.9476 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  18.81460    1.85273   10.15  < 2e-16 ***
## anxiety      -4.97617    0.80918   -6.15 1.73e-08 ***
## I(anxiety^2)  1.02500    0.07529   13.62  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.08 on 97 degrees of freedom
## Multiple R-squared:  0.9221, Adjusted R-squared:  0.9205 
## F-statistic: 574.3 on 2 and 97 DF,  p-value: < 2.2e-16

3. Both positive