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
-1	7
0	10
1	10
2	8

help me solve this view an example get more help - clear all final check

Explanation:

Step1: Calculate predicted values

For each $x$, compute $\hat{y}=f(x)=-x^2+2x+4$:

• $x=-2$: $\hat{y}=-(-2)^2+2(-2)+4=-4-4+4=-4$
• $x=-1$: $\hat{y}=-(-1)^2+2(-1)+4=-1-2+4=1$
• $x=0$: $\hat{y}=-(0)^2+2(0)+4=4$
• $x=1$: $\hat{y}=-(1)^2+2(1)+4=-1+2+4=5$
• $x=2$: $\hat{y}=-(2)^2+2(2)+4=-4+4+4=4$

Step2: Match data to scatterplot

Plot data points $(-2,0), (-1,7), (0,10), (1,10), (2,8)$ and the parabola $y=-x^2+2x+4$. The leftmost grid matches these points and the curve.

Step3: Compute residuals

Residual = $y-\hat{y}$:

• $x=-2$: $0-(-4)=4$
• $x=-1$: $7-1=6$
• $x=0$: $10-4=6$
• $x=1$: $10-5=5$
• $x=2$: $8-4=4$

Step4: Evaluate fit

Residuals are large positive values, meaning observed $y$-values are much higher than predicted values.[LLM SSE On Failure]

Answer:

1. Correct scatterplot + function graph: First grid (leftmost)
2. Residuals: $(-2): 0 - (-(-2)^2 + 2(-2)+4) = 0 - (-4-4+4)=4$; $(-1):7 - (-(-1)^2+2(-1)+4)=7 - (-1-2+4)=6$; $(0):10 - (0+0+4)=6$; $(1):10 - (-1+2+4)=5$; $(2):8 - (-4+4+4)=4$
3. Fit description: Poor fit, residuals are large and positive