Question

question 10 of 10
what is the exponential regression equation that fits these data?

x	y
-3	0.075
-2	0.25
-1	0.70
0	3
1	8
2	20
3	60
4	160

a. $y=2.01 \cdot 3.80^{x}$
b. $y=2.26 \cdot 3.02^{x}$
c. $y=3.02 \cdot 2.26^{x}$
d. $y=4.90x^{2}+14.44x - 4.69$

Explanation:

Step1: Recall exponential form

Exponential regression has the form $y = ab^x$, where when $x=0$, $y=a$.

Step2: Find a using x=0

From the table, when $x=0$, $y=3$. So $a \approx 3$.

Step3: Test x=1 for best b

For $x=1$, $y=8$. Using $a \approx 3$, solve $8 = 3b^1$, so $b \approx \frac{8}{3} \approx 2.67$. Now test options:

• Option A: $2.01 \cdot 3.80^1 = 7.638$
• Option B: $2.26 \cdot 3.02^1 \approx 6.825$
• Option C: $3.02 \cdot 2.26^1 \approx 6.825$

Wait, test x=2: $y=20$

• Option A: $2.01 \cdot 3.80^2 = 2.01 \cdot 14.44 = 29.0244$
• Option B: $2.26 \cdot 3.02^2 \approx 2.26 \cdot 9.1204 \approx 20.61$
• Option C: $3.02 \cdot 2.26^2 \approx 3.02 \cdot 5.1076 \approx 15.42$

Test x=3: $y=60$

• Option B: $2.26 \cdot 3.02^3 \approx 2.26 \cdot 27.5436 \approx 62.25$
• Option A: $2.01 \cdot 3.80^3 = 2.01 \cdot 54.872 = 110.29$
• Option C: $3.02 \cdot 2.26^3 \approx 3.02 \cdot 11.543 \approx 34.86$

Test x=-1: $y=0.70$

• Option B: $2.26 \cdot 3.02^{-1} \approx 2.26 \cdot 0.331 \approx 0.748$
• Option A: $2.01 \cdot 3.80^{-1} \approx 2.01 \cdot 0.263 \approx 0.529$
• Option C: $3.02 \cdot 2.26^{-1} \approx 3.02 \cdot 0.442 \approx 1.335$

Option B's values are closest to the table data, and D is quadratic, not exponential.

Answer:

B. $y=2.26 \cdot 3.02^{x}$