Math Problem Statement
Determine if the data in the table of values shows exponential growth, exponential decay, or neither.
Solution
Let's analyze the given data table to determine if it shows exponential growth, exponential decay, or neither.
The table provided is:
\hline x & 2 & 3 & 4 & 5 & 6 \\ \hline y & 7,776 & 1,296 & 216 & 36 & 6 \\ \hline \end{array}$$ ### To check for exponential behavior, let's look for a constant ratio between consecutive $$y$$-values. - From $$y_2 = 7,776$$ to $$y_3 = 1,296$$: $$\frac{y_3}{y_2} = \frac{1,296}{7,776} = \frac{1}{6}$$ - From $$y_3 = 1,296$$ to $$y_4 = 216$$: $$\frac{y_4}{y_3} = \frac{216}{1,296} = \frac{1}{6}$$ - From $$y_4 = 216$$ to $$y_5 = 36$$: $$\frac{y_5}{y_4} = \frac{36}{216} = \frac{1}{6}$$ - From $$y_5 = 36$$ to $$y_6 = 6$$: $$\frac{y_6}{y_5} = \frac{6}{36} = \frac{1}{6}$$ ### Conclusion: Since the ratio between consecutive $$y$$-values is consistently $$\frac{1}{6}$$, the data exhibits **exponential decay**. Would you like more details on how exponential decay is determined, or do you have any other questions? ### Related Questions: 1. How do you differentiate between exponential growth and decay? 2. What is the general form of an exponential decay function? 3. Can exponential decay apply to real-world phenomena like radioactive decay? 4. How does exponential decay compare with linear decay? 5. What happens to $$y$$ values as $$x$$ increases in an exponential decay function? ### Tip: In exponential decay, the common ratio between consecutive values is always less than 1, which causes the values to decrease rapidly over time.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Exponential Growth
Exponential Decay
Ratios
Sequences
Formulas
y = ab^x, where 0 < b < 1 for decay
Theorems
Exponential Function Theorem
Suitable Grade Level
Grades 8-10