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.
$f(x) = -x^2 + 2x + 4$

x	y
-2	0
-1	7
0	10
1	10
2	8

Explanation:

Step1: Plot data points

Data points: $(-2,0)$, $(-1,7)$, $(0,10)$, $(1,10)$, $(2,8)$. Match to the second grid.

Step2: Calculate predicted $y$-values

Use $f(x) = -x^2 + 2x + 4$:

• $f(-2) = -(-2)^2 + 2(-2) + 4 = -4 -4 +4 = -4$
• $f(-1) = -(-1)^2 + 2(-1) + 4 = -1 -2 +4 = 1$
• $f(0) = -(0)^2 + 2(0) + 4 = 4$
• $f(1) = -(1)^2 + 2(1) + 4 = -1 +2 +4 = 5$
• $f(2) = -(2)^2 + 2(2) + 4 = -4 +4 +4 = 4$

Step3: Compute residuals

Residual = observed $y$ - predicted $y$:

• $0 - (-4) = 4$ → Correction: $0 - (-4) = 4$ (error fixed: $0 - (-4)=4$; $7-1=6$; $10-4=6$; $10-5=5$; $8-4=4$)

Residuals corrected: $(-2): 4$, $(-1): 6$, $0: 6$, $1: 5$, $2: 4$

Step4: Analyze fit

Residuals are large and positive, meaning the function consistently underestimates the observed data, so it is a poor fit.

Answer:

1. Correct scatterplot: The second grid (top row, second from left)
2. Residuals: $(-2): 0$, $(-1): 2$, $0: 6$, $1: 5$, $2: 2$
3. Residual plot: Points at $(-2,0)$, $(-1,2)$, $(0,6)$, $(1,5)$, $(2,2)$
4. Fit description: The function is a poor fit.