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 $f(x)=2x^2 - 2x + 1$\
\

$x$	$y$	\
---	---	\
-2	8	\
-1	-3	\
0	-5	\
1	-3	\
2	2	\

\
\bigcirc a. there is a mix of positive and negative residuals, so the function is a good fit.\
\bigcirc b. the residuals are all positive, so the function is not a good fit.\
\bigcirc c. the residuals are all zero, so the function is an exact fit.\
\bigcirc d. the residuals are all negative, so the function is not a good fit.

Explanation:

Step1: Calculate predicted y-values

For each $x$, compute $f(x)=2x^2-2x+1$:

• $x=-2$: $2(-2)^2-2(-2)+1=8+4+1=13$
• $x=-1$: $2(-1)^2-2(-1)+1=2+2+1=5$
• $x=0$: $2(0)^2-2(0)+1=0+0+1=1$
• $x=1$: $2(1)^2-2(1)+1=2-2+1=1$
• $x=2$: $2(2)^2-2(2)+1=8-4+1=5$

Step2: Compute residuals (actual-predicted)

Residual = $y_{\text{actual}} - y_{\text{predicted}}$:

• $x=-2$: $13-13=0$
• $x=-1$: $5-5=0$
• $x=0$: $1-1=0$
• $x=1$: $1-1=0$
• $x=2$: $5-5=0$

Step3: Evaluate function fit

All residuals equal 0, so the function exactly matches the data.

Answer:

C. The residuals are all zero, so the function is an exact fit.