Question

which table represents exponential growth? (four tables with x and y values are shown: first table: x=1,y=2; x=2,y=4; x=3,y=8; x=4,y=16. second table: x=1,y=2; x=2,y=4; x=3,y=7; x=4,y=11. third table: x=1,y=2; x=2,y=4; x=3,y=6; x=4,y=8. fourth table: x=1,y=2; x=2,y=4; x=3,y=6; x=4,y=10.)

Explanation:

Step1: Recall exponential growth

Exponential growth has a constant ratio between consecutive \( y \)-values (when \( x \) increases by 1). The formula is \( y = ab^{x} \), so \( \frac{y_{n + 1}}{y_{n}}=b \) (constant).

Step2: Analyze first table

For \( x = 1, y = 2 \); \( x = 2, y = 4 \); \( x = 3, y = 8 \); \( x = 4, y = 16 \).
Calculate ratios: \( \frac{4}{2}=2 \), \( \frac{8}{4}=2 \), \( \frac{16}{8}=2 \). Ratio is constant (\( b = 2 \)).

Step3: Analyze second table

\( x = 1, y = 2 \); \( x = 2, y = 4 \); \( x = 3, y = 7 \); \( x = 4, y = 11 \).
Ratios: \( \frac{4}{2}=2 \), \( \frac{7}{4}=1.75 \), \( \frac{11}{7}\approx1.57 \). Not constant.

Step4: Analyze third table

\( x = 1, y = 2 \); \( x = 2, y = 4 \); \( x = 3, y = 6 \); \( x = 4, y = 8 \).
Ratios: \( \frac{4}{2}=2 \), \( \frac{6}{4}=1.5 \), \( \frac{8}{6}\approx1.33 \). Not constant.

Step5: Analyze fourth table

\( x = 1, y = 2 \); \( x = 2, y = 4 \); \( x = 3, y = 6 \); \( x = 4, y = 10 \).
Ratios: \( \frac{4}{2}=2 \), \( \frac{6}{4}=1.5 \), \( \frac{10}{6}\approx1.67 \). Not constant.

Answer:

The table in the top - left (with \( x = 1,y = 2 \); \( x = 2,y = 4 \); \( x = 3,y = 8 \); \( x = 4,y = 16 \)) represents exponential growth.