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Part 1
Devin found the equation of the parabola that fits the three points in the table to be y equals 0.345 x squared minus 0.57 x minus 2.78. Is Devin​ correct? Explain.
x
minus4
0.6
9
y
5
minus3
20
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer boxes within your choice.
A.
No. This is close to the​ answer, but it is never acceptable to round the values in a model. Since an exact answer can be found with 3​ points, Devin should have reported aequals

enter your response here​, bequals

enter your response here​, and cequals

enter your response here.
​(Type an exact​ answer.)
B.
Yes. Even though​ Devin's equation does not give the points listed in the​ table, that is only because Devin rounded the values of​ a, b, and c appropriately.
C.
Yes.​ Devin's equation gives exactly the points in the table.
D.
No. This is close to the​ answer, but when the exact model is found by substituting the 3 points into the standard form and​ solving, the parameters are aequals

enter your response here​, bequals

enter your response here​, and cequals

enter your response here.
​(Type an integer or decimal rounded to two decimal places as​ needed.)
Explain.Explain.Explain.Explain.Explain.Explain.Select the correct choice below and, if necessary, fill in the answer boxes within your choice.
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To determine whether Devin's equation is correct, we need to check if the given quadratic equation \( y = 0.345x^2 - 0.57x - 2.78 \) fits the points in the table. The points provided are \((-4, 5)\), \((0.6, -3)\), and \((9, 20)\).

### Step 1: Substitute each point into the equation
We substitute the \( x \) values into the equation to see if the corresponding \( y \) values match.

1. **For \( x = -4 \) and \( y = 5 \):**

\[
y = 0.345(-4)^2 - 0.57(-4) - 2.78
\]
\[
y = 0.345(16) + 2.28 - 2.78
\]
\[
y = 5.52 + 2.28 - 2.78 = 5.02 \quad (\text{close to } 5)
\]

2. **For \( x = 0.6 \) and \( y = -3 \):**

\[
y = 0.345(0.6)^2 - 0.57(0.6) - 2.78
\]
\[
y = 0.345(0.36) - 0.342 - 2.78
\]
\[
y = 0.1242 - 0.342 - 2.78 = -2.9978 \quad (\text{close to } -3)
\]

3. **For \( x = 9 \) and \( y = 20 \):**

\[
y = 0.345(9)^2 - 0.57(9) - 2.78
\]
\[
y = 0.345(81) - 5.13 - 2.78
\]
\[
y = 27.945 - 5.13 - 2.78 = 20.035 \quad (\text{close to } 20)
\]

### Step 2: Analyze the Options
- **Option A** suggests that rounding is not acceptable and implies that Devin should have used exact values for \(a\), \(b\), and \(c\). However, this option is not fully correct without verifying if exact values yield the original points.
  
- **Option B** states that Devin rounded the values of \(a\), \(b\), and \(c\) appropriately, but the equation does not give the exact points.
  
- **Option C** claims Devin's equation gives exactly the points, which isn't true since the points are close but not exact.
  
- **Option D** indicates that the exact model parameters should be calculated by solving the system of equations formed by substituting the points into the general form of a quadratic equation. This would be more accurate.

### Conclusion
The most appropriate answer is **Option D**, since it acknowledges that Devin's equation is close, but exact parameters should be calculated. When substituting the points into the standard quadratic form and solving, the exact coefficients \(a\), \(b\), and \(c\) would differ slightly.

Would you like to see the detailed calculations for solving the exact quadratic equation using these points?

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Here are some related questions you might be interested in:

1. How do you derive the exact coefficients \(a\), \(b\), and \(c\) for a quadratic equation?
2. Why is it important to avoid rounding intermediate steps when finding a model?
3. How does the choice of points affect the accuracy of the quadratic model?
4. Can a quadratic model exactly fit any three non-collinear points?
5. What are some common pitfalls when approximating a mathematical model?

**Tip:** When modeling data, always perform a residual analysis to understand how well your model fits the data points.

Analyze whether Devin's quadratic equation fits the points (-4, 5), (0.6, -3), and (9, 20) using step-by-step substitution and evaluation. Understand the implications of rounding coefficients in mathematical modeling. Suitable for high school students studying quadratic equations.

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