Math Problem Statement
Look at this table: x y – 5 2187 320 – 4 729 160 – 3 243 80 – 2 81 40 – 1 27 20 Write a linear function (y=mx+b) or an exponential function (y=a(b)x) that models the data.
Solution
To determine the best model for this data, we will first analyze whether a linear function or an exponential function fits the data.
Data:
\hline x & y \\ \hline -5 & 2187 \\ -4 & 729 \\ -3 & 243 \\ -2 & 81 \\ -1 & 27 \\ \hline \end{array}$$ ### Step 1: Check if an exponential model works. An exponential function has the form $$y = a(b)^x$$, where: - $$a$$ is the initial value (when $$x = 0$$), - $$b$$ is the base (growth or decay factor). We notice that as $$x$$ decreases by 1, $$y$$ seems to be divided by 3. For example: - When $$x = -4$$, $$y = 729$$, - When $$x = -3$$, $$y = 243$$, - $$729 / 243 = 3$$. This suggests that $$b = \frac{1}{3}$$, indicating a decay factor of 1/3. ### Step 2: Find the initial value $$a$$. We know that the general form is $$y = a \left( \frac{1}{3} \right)^x$$. Let's use one of the points to find $$a$$. For instance, when $$x = -5$$, $$y = 2187$$, so: $$2187 = a \left( \frac{1}{3} \right)^{-5}$$ Since $$\left( \frac{1}{3} \right)^{-5} = 3^5 = 243$$, we have: $$2187 = a \times 243$$ Solving for $$a$$: $$a = \frac{2187}{243} = 9$$ ### Step 3: Write the exponential function. Thus, the exponential function that models the data is: $$y = 9 \left( \frac{1}{3} \right)^x$$ ### Final Answer: The exponential function is: $$y = 9 \left( \frac{1}{3} \right)^x$$ Would you like further clarification or assistance with other questions? Here are some related questions to explore: 1. How do we find the exponential decay rate for different data points? 2. What happens if the base of an exponential function is greater than 1? 3. How do we interpret the value of $$a$$ in an exponential function? 4. How would we determine if a linear function might be a better fit for this data? 5. How does the exponential model change when the sign of $$x$$ changes? **Tip**: Always look for consistent multiplicative changes in the data when considering an exponential function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Decay Models
Algebra
Linear vs. Exponential Functions
Formulas
Exponential function: y = a(b)^x
Theorems
Exponential decay model
Suitable Grade Level
Grades 8-10
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