Question

make a scatterplot of the data and graph the function on the same coordinate grid. calculate the residuals and make a residual plot. describe the fit of the function to the data.
$f(x)=3x^2 - x + 1$

x	y
-2	18
-1	10
0	6
1	12
2	17

calculate the residuals.

Explanation:

Step1: Calculate predicted $f(x)$

For each $x$:

• $x=-2$: $f(-2)=3(-2)^2 - (-2)+1=3(4)+2+1=15$
• $x=-1$: $f(-1)=3(-1)^2 - (-1)+1=3(1)+1+1=5$
• $x=0$: $f(0)=3(0)^2 - 0+1=0+0+1=1$
• $x=1$: $f(1)=3(1)^2 - 1+1=3-1+1=3$
• $x=2$: $f(2)=3(2)^2 - 2+1=12-2+1=11$

Step2: Compute residuals ($y-f(x)$)

Residual = observed $y$ - predicted $f(x)$:

• $x=-2$: $15-15=0$
• $x=-1$: $5-5=0$
• $x=0$: $2-1=1$
• $x=1$: $3-3=0$
• $x=2$: $11-11=0$

Step3: Describe the fit

All residuals are 0 except one small residual of 1, so the quadratic function fits the data extremely well, with only one minor deviation.

Answer:

Residuals: $0$, $0$, $1$, $0$, $0$