Question

question 7 of 13 which of the following quadratic regression equations best fits the data shown below? x -4 -3 -2 -1 0 1 2 3 4 y 60 34 10 6 8 14 20 56 98 a. y = 2.87x² + 2.70x + 4.07 b. y = 2.09x² + 1.30x + 4.61 c. y = 3.11x² + 1.20x + 3.64 d. y = 4.67x² + 4.10x + 2.87

Explanation:

Step1: Substitute \(x = 0\) into each equation

When \(x = 0\), for option A: \(y=2.87\times0^{2}+2.70\times0 + 4.07=4.07\); for option B: \(y=2.09\times0^{2}+1.30\times0 + 4.61 = 4.61\); for option C: \(y=3.11\times0^{2}+1.20\times0+3.64 = 3.64\); for option D: \(y=4.67\times0^{2}+4.10\times0 + 2.87=2.87\). From the data table, when \(x = 0\), \(y = 8\). This step helps to eliminate some options quickly. Since none match exactly, we use another \(x\) - value.

Step2: Substitute \(x = 1\) into each equation

For option A: \(y=2.87\times1^{2}+2.70\times1+4.07=2.87 + 2.70+4.07=9.64\); for option B: \(y=2.09\times1^{2}+1.30\times1 + 4.61=2.09+1.30 + 4.61=7.99\); for option C: \(y=3.11\times1^{2}+1.20\times1+3.64=3.11+1.20 + 3.64=7.95\); for option D: \(y=4.67\times1^{2}+4.10\times1+2.87=4.67+4.10+2.87 = 11.64\). From the data - table, when \(x = 1\), \(y = 14\). This is also not conclusive.

Step3: Substitute \(x=-1\) into each equation

For option A: \(y=2.87\times(-1)^{2}+2.70\times(-1)+4.07=2.87-2.70 + 4.07=4.24\); for option B: \(y=2.09\times(-1)^{2}+1.30\times(-1)+4.61=2.09-1.30 + 4.61=5.4\); for option C: \(y=3.11\times(-1)^{2}+1.20\times(-1)+3.64=3.11-1.20+3.64 = 5.55\); for option D: \(y=4.67\times(-1)^{2}+4.10\times(-1)+2.87=4.67-4.10+2.87=3.44\). From the data - table, when \(x=-1\), \(y = 6\).

Step4: Substitute \(x = 2\) into each equation

For option A: \(y=2.87\times2^{2}+2.70\times2+4.07=2.87\times4 + 5.4+4.07=11.48+5.4+4.07=20.95\); for option B: \(y=2.09\times2^{2}+1.30\times2+4.61=2.09\times4+2.6 + 4.61=8.36+2.6+4.61 = 15.57\); for option C: \(y=3.11\times2^{2}+1.20\times2+3.64=3.11\times4+2.4+3.64=12.44+2.4+3.64 = 18.48\); for option D: \(y=4.67\times2^{2}+4.10\times2+2.87=4.67\times4+8.2+2.87=18.68+8.2+2.87=29.75\). From the data - table, when \(x = 2\), \(y = 20\). Option C is the closest.

Answer:

C. \(y = 3.11x^{2}+1.20x + 3.64\)