Math Problem Statement
The table of ordered pairs (x, y) gives an exponential function. Write an equation for the function. The table is as follows:
x | y
-1 | 40
0 | 20
1 | 10
2 | 5
Solution
The table provided suggests an exponential function of the form:
where is the initial value (when ), and is the growth/decay factor. Let's analyze the table:
\hline x & y \\ \hline -1 & 40 \\ 0 & 20 \\ 1 & 10 \\ 2 & 5 \\ \hline \end{array}$$ 1. When $$x = 0$$, $$y = 20$$, so $$a = 20$$. 2. To find $$b$$, note how the $$y$$-values change as $$x$$ increases. Each time $$x$$ increases by 1, $$y$$ is halved (from 40 to 20, 20 to 10, and 10 to 5). This indicates that $$b = \frac{1}{2}$$. Thus, the equation for the exponential function is: $$y = 20 \cdot \left(\frac{1}{2}\right)^x$$ Would you like more details or clarifications? Here are 5 related questions: 1. How do we identify the base $$b$$ in exponential functions from a table of values? 2. What happens if the base $$b$$ is greater than 1? 3. How can we solve for $$x$$ when given a specific $$y$$-value in an exponential equation? 4. How do exponential growth and decay differ in form and behavior? 5. Can we find the inverse of an exponential function like the one we derived? **Tip:** Always check if the ratio between consecutive $$y$$-values is consistent; this helps confirm the exponential nature of the function!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Patterns in Sequences
Algebra
Formulas
y = a * b^x
Theorems
Exponential Growth/Decay Theorem
Suitable Grade Level
Grades 8-10