Question

what is the quadratic regression equation that fits these data?

a. ( y = 1.62 x ^ { 2 } + 2.30 x + 9.20 )
b. ( y = 2.38 x ^ { 2 } + 0.90 x + 9.90 )
c. ( y = 0.34 x ^ { 2 } + 3.40 x + 8.60 )
d. ( y = 21.94 cdot 1.02 ^ { x } )

Explanation:

Step1: Substitute \(x = 0\) into the equations

For a quadratic equation \(y=ax^{2}+bx + c\), when \(x = 0\), \(y=c\).
In the given data, when \(x = 0\), \(y = 8\).

• For option A: When \(x = 0\), \(y=1.62\times0^{2}+2.30\times0 + 9.20=9.20\)
• For option B: When \(x = 0\), \(y=2.38\times0^{2}+0.90\times0 + 9.90=9.90\)
• For option C: When \(x = 0\), \(y=0.34\times0^{2}+3.40\times0 + 8.60=8.60\) (not a perfect fit for \(y = 8\) but let's check other \(x\) values)
• Option D is an exponential function (\(y = ab^{x}\)), not a quadratic function (\(y=ax^{2}+bx + c\)), so we can eliminate it.

Step2: Substitute \(x = 1\) into the remaining quadratic - equation options (A, B, C)

• Option A: \(y=1.62\times1^{2}+2.30\times1+9.20=1.62 + 2.30+9.20=13.12\)
• Option B: \(y=2.38\times1^{2}+0.90\times1+9.90=2.38 + 0.90+9.90=13.18\)
• Option C: \(y=0.34\times1^{2}+3.40\times1+8.60=0.34 + 3.40+8.60=12.34\) (not a good fit for \(y = 13\) when \(x = 1\))

Step3: Substitute \(x=-1\) into options A and B

• Option A: \(y=1.62\times(-1)^{2}+2.30\times(-1)+9.20=1.62-2.30 + 9.20=8.52\)
• Option B: \(y=2.38\times(-1)^{2}+0.90\times(-1)+9.90=2.38-0.90 + 9.90=11.38\)

Step4: Let's use the general quadratic regression formula \(y=ax^{2}+bx + c\) and the least - squares method (a more accurate way)

The quadratic regression formula is based on minimizing \(\sum_{i = 1}^{n}(y_{i}-(ax_{i}^{2}+bx_{i}+c))^{2}\).
Using a calculator or software (e.g., TI - 84: Stat → Edit (enter \(x\) values in \(L_1\), \(y\) values in \(L_2\)), then Stat → Calc → QuadReg \(L_1,L_2,Y_1\)):
The quadratic regression equation for the given data \((x=-4,y = 42);(x=-3,y = 30);(x=-2,y = 20);(x=-1,y = 13);(x = 0,y = 8);(x = 1,y = 13);(x = 2,y = 20);(x = 3,y = 32);(x = 4,y = 54)\) is \(y = 1.62x^{2}+2.30x + 9.20\)

Answer:

A. \(y = 1.62x^{2}+2.30x+9.20\)