Question

which table shows exponential decay?
(then four tables with x and y values are shown:
first table: x:1,2,3,4; y:16,12,8,4
second table: x:1,2,3,4; y:16,12,9,7
third table: x:1,2,3,4; y:16,8,3,1
fourth table: x:1,2,3,4; y:16,8,4,2)

Explanation:

Step1: Recall Exponential Decay

Exponential decay has a constant ratio between consecutive \( y \)-values (multiplicative decay), while linear decay has a constant difference (additive decay).

Step2: Analyze First Table

• \( x=1,y=16 \); \( x=2,y=12 \); \( x=3,y=8 \); \( x=4,y=4 \)
• Differences: \( 16 - 12 = 4 \), \( 12 - 8 = 4 \), \( 8 - 4 = 4 \) (constant difference, linear decay)
• Ratios: \( \frac{12}{16}=\frac{3}{4} \), \( \frac{8}{12}=\frac{2}{3} \) (not constant)

Step3: Analyze Second Table

• \( x=1,y=16 \); \( x=2,y=12 \); \( x=3,y=9 \); \( x=4,y=7 \)
• Differences: \( 16 - 12 = 4 \), \( 12 - 9 = 3 \), \( 9 - 7 = 2 \) (not constant)
• Ratios: \( \frac{12}{16}=\frac{3}{4} \), \( \frac{9}{12}=\frac{3}{4} \), \( \frac{7}{9}\approx0.78 \) (not constant after third term)

Step4: Analyze Third Table

• \( x=1,y=16 \); \( x=2,y=8 \); \( x=3,y=3 \); \( x=4,y=1 \)
• Differences: \( 16 - 8 = 8 \), \( 8 - 3 = 5 \), \( 3 - 1 = 2 \) (not constant)
• Ratios: \( \frac{8}{16}=\frac{1}{2} \), \( \frac{3}{8}=0.375 \), \( \frac{1}{3}\approx0.33 \) (not constant)

Step5: Analyze Fourth Table

• \( x=1,y=16 \); \( x=2,y=8 \); \( x=3,y=4 \); \( x=4,y=2 \)
• Differences: \( 16 - 8 = 8 \), \( 8 - 4 = 4 \), \( 4 - 2 = 2 \) (not constant)
• Ratios: \( \frac{8}{16}=\frac{1}{2} \), \( \frac{4}{8}=\frac{1}{2} \), \( \frac{2}{4}=\frac{1}{2} \) (constant ratio \( \frac{1}{2} \), exponential decay)

Answer:

The table with \( x \) values 1, 2, 3, 4 and \( y \) values 16, 8, 4, 2 (the bottom - left table)