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 + 8 )

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

make a scatterplot of the data and graph the function on the same coordinate grid. choose the correct answer below.

a.

b.

c.

d.

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Explanation:

Step1: Calculate function values

For each $x$, compute $\hat{y}=-x^2+2x+8$:

• $x=-2$: $\hat{y}=-(-2)^2+2(-2)+8=-4-4+8=0$
• $x=-1$: $\hat{y}=-(-1)^2+2(-1)+8=-1-2+8=7$
• $x=0$: $\hat{y}=-(0)^2+2(0)+8=8$
• $x=1$: $\hat{y}=-(1)^2+2(1)+8=-1+2+8=9$
• $x=2$: $\hat{y}=-(2)^2+2(2)+8=-4+4+8=8$

Step2: Compute residuals

Residual = $y-\hat{y}$ (actual - predicted):

• $x=-2$: $0-0=0$
• $x=-1$: $7-7=0$
• $x=0$: $10-8=2$
• $x=1$: $10-9=1$
• $x=2$: $8-8=0$

Step3: Assess function fit

Residuals are small (0, 0, 2, 1, 0) with no clear pattern, meaning the function aligns closely with the data points, so it is a very good fit.
To select the correct plot: confirm the data points are plotted, and the parab[LLM SSE On Failure]

Answer:

First, we identify the correct scatterplot + function graph (note: since the option images are not fully visible, we first calculate the function values, then residuals, then assess fit):

1. Correct plot will have data points (-2,0), (-1,7), (0,10), (1,10), (2,8) and the curve $y=-x^2+2x+8$
2. Residuals: $-1, 0, 2, 1, 0$
3. The function is a very good fit to the data.