Math Problem Statement
Complete the equation for the exponential function h(x) represented in the table, with given values h(0) = 10 and h(1) = 4.
Solution
To solve for the exponential function based on the given table values:
The table provided:
x & h(x) \\ \hline 0 & 10 \\ 1 & 4 \\ \end{array}$$ 1. **Determine $$a$$:** When $$x = 0$$, $$h(0) = a \cdot b^0 = a \cdot 1 = a$$. - Since $$h(0) = 10$$, we find $$a = 10$$. 2. **Solve for $$b$$:** When $$x = 1$$, we have $$h(1) = a \cdot b = 10 \cdot b$$. - Given $$h(1) = 4$$, substitute $$a = 10$$: $$10 \cdot b = 4$$ - Solving for $$b$$: $$b = \frac{4}{10} = 0.4$$ 3. **Construct the equation:** With $$a = 10$$ and $$b = 0.4$$, the exponential function becomes: $$h(x) = 10 \cdot (0.4)^x$$ Thus, the completed equation for $$h(x)$$ is: $$h(x) = 10 \cdot (0.4)^x$$ Would you like further details on exponential functions or have any questions? Here are five questions that might expand your understanding: 1. How would the function change if the value of $$h(0)$$ was different? 2. What happens to the graph of $$h(x)$$ as $$b$$ approaches 1? 3. How does an exponential function differ from a linear function in terms of growth or decay? 4. Can you determine the rate of decay for this function, given that $$b = 0.4$$? 5. What are the effects of changing the base $$b$$ in exponential functions? **Tip:** For exponential functions, the base $$b$$ determines whether the function represents growth ($$b > 1$$) or decay ($$0 < b < 1$$).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential Functions
Formulas
Exponential function formula h(x) = a * b^x
Theorems
Properties of exponential functions
Suitable Grade Level
Grades 9-10