Math Problem Statement
Find the equation of the exponential function represented by the table below: x | y
0 1
1 4
2 16
3 64
Solution
The table provided in the image shows the following data for and :
\hline x & y \\ \hline 0 & 1 \\ 1 & 4 \\ 2 & 16 \\ 3 & 64 \\ \hline \end{array}$$ We are asked to find the equation of the exponential function that fits this data. An exponential function generally takes the form: $$y = ab^x$$ Where $$a$$ is the initial value (when $$x = 0$$), and $$b$$ is the base of the exponential function. ### Step 1: Find $$a$$ From the table, when $$x = 0$$, $$y = 1$$. This gives us: $$y = ab^0 = a \quad \Rightarrow \quad a = 1$$ ### Step 2: Find $$b$$ We can use any other data point from the table to find $$b$$. Using the point $$(1, 4)$$, we substitute into the equation: $$y = ab^x \quad \Rightarrow \quad 4 = 1 \cdot b^1$$ So, $$b = 4$$. ### Step 3: Write the equation Now that we know $$a = 1$$ and $$b = 4$$, the equation becomes: $$y = 4^x$$ This is the equation of the exponential function that represents the data in the table. Would you like further clarification or assistance with other related topics? Here are some questions you might find helpful: 1. How do exponential functions differ from linear functions? 2. How can I verify the correctness of this exponential equation? 3. What are some real-world examples of exponential growth or decay? 4. What would happen if the table values had been different (e.g., $$y$$ values decreasing)? 5. How do you calculate the base $$b$$ if the equation is in the form $$y = ab^x$$ but without an explicit value for $$a$$? **Tip**: When working with exponential equations, always check the behavior of the values as they change. If the data grows rapidly, like in this case, an exponential function is likely.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Exponential Functions
Algebra
Formulas
y = ab^x
Theorems
Exponential Function Properties
Suitable Grade Level
Grades 7-9