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.
$f(x)=3x^2 - 2x + 1$

x	y
-2	8
-1	3
0	-6
1	-2
2	2

Explanation:

Step1: Identify data points

Data points: $(-2,8), (-1,-3), (0,-6), (1,-3), (2,2)$

Step2: Calculate predicted values

For each $x$, compute $\hat{y}=2x^2-2x-6$:

• $x=-2$: $\hat{y}=2(-2)^2-2(-2)-6=8+4-6=8$
• $x=-1$: $\hat{y}=2(-1)^2-2(-1)-6=2+2-6=-3$
• $x=0$: $\hat{y}=2(0)^2-2(0)-6=0-0-6=-6$
• $x=1$: $\hat{y}=2(1)^2-2(1)-6=2-2-6=-3$
• $x=2$: $\hat{y}=2(2)^2-2(2)-6=8-4-6=2$

Step3: Compute residuals

Residual = $y-\hat{y}$ for each point:

• $8-8=0$, $-3-(-3)=0$, $-6-(-6)=0$, $-3-(-3)=0$, $2-2=0$

Step4: Evaluate function fit

All residuals are 0, so every data point lies exactly on the function.

Step5: Match to scatterplot

The middle-left graph plots all given data points correctly with the parabola passing through each point.

Answer:

1. Correct scatterplot + function graph: The second graph (middle-left one with points (-2,8), (-1,-3), (0,-6), (1,-3), (2,2) plotted with the parabola $f(x)=2x^2-2x-6$)
2. Residuals: $0, 0, 0, 0, 0$
3. Residual plot: A horizontal line of points on the x-axis at x=-2,-1,0,1,2
4. Fit description: Perfect fit