Question

Use the data set to create a quadratic function if it applies. Use the model to predict the value of x when y = -4.

(-3, 10), (0, 4), (3, -1), (6, -5), (9, -8)


Question 3 options:

A)

29.32 and 3.97


B)

32.27 and 4.57


C)

27.48 and 5.20


D)

No Solution

Answer 1

The closest choice is ...

C) 27.48 and 5.20

Explanation:

Using the equation ...

y = ax² + bx + c

You can substitute the first three points to get linear equations in a, b, and c.

10 = a(-3)² +b(-3) +c = 9a -3b +c

4 = a(0)² + b(0) +c = c

-1 = a(3)² + b(3) +c = 9a +3b +c

The second equation tells us c=4. Subtracting the first equation from the last, we get ...

(9a +3b +c) -(9a -3b +c) = (-1) -(10)

6b = -11

b = -11/6

Then we can find "a" from the first equation:

10 = 9a -3(-11/6) +4

1/2 = 9a . . . . subtract 19/2

1/18 = a . . . . . divide by 9

The model is ...

y = (1/18)x² -(11/6)x +4 . . . . . . . . [call this "model A"]

y = (x² -33x +72)/18

Solving this for y = -4, we have ...

-4 = (x² -33x +72)/18

x² -33x +144 = 0 . . . . . . multiply by 18, add 72

(x -16.5)² -128.25 = 0 . . . . put in vertex form

x = 16.5 ±√128.25 ≈ {5.17525, 27.8248}

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The closest values to these among the choices offered are those of choice C. It appears that the coefficients of model A were rounded to 3 decimal places (or so) before the predicted x-values were computed.