Math Problem Statement
Analyze the exponential function based on the table of x and y values, identify whether it represents growth or decay, and complete the statements.
Solution
The provided content involves analyzing an exponential function based on a table of and values, and completing statements about exponential growth or decay. Let’s work through it step by step:
Analysis of the Table
From the first image, the and values in the table are:
\hline x & y \\ \hline 0 & 4 \\ 0.5 & 5.1381 \\ 1 & 6.6 \\ 2.5 & 13.988 \\ 3 & 17.969 \\ 5 & 48.919 \\ \hline \end{array}$$ 1. **Observing Growth Behavior**: - As $$x$$ increases, $$y$$ increases. The rate of increase suggests **exponential growth** because the values grow at an increasing rate. 2. **General Form of Exponential Equations**: - The equation for exponential functions is: $$y = a(b)^x$$ Where: - $$a$$: Initial value (when $$x = 0$$), - $$b$$: Growth factor (if $$b > 1$$, it’s growth; if $$0 < b < 1$$, it’s decay). 3. **Initial Value $$a$$**: - From the table, when $$x = 0$$, $$y = 4$$. So, $$a = 4$$. 4. **Growth Factor $$b$$**: - To find $$b$$, use any two points in the table. For instance: - When $$x = 0$$, $$y = 4$$, - When $$x = 1$$, $$y = 6.6$$. Substituting into the exponential equation: $$6.6 = 4(b)^1 \quad \Rightarrow \quad b = \frac{6.6}{4} = 1.65$$ - Hence, $$b = 1.65$$, indicating exponential growth. 5. **Equation**: - The equation is: $$y = 4(1.65)^x$$ --- ### **Completing the Statements** From the second image: 1. The table represents **exponential growth** because $$b > 1$$. 2. The table has values of $$x$$ that are **increasing**, while values of $$y$$ are **increasing**. 3. This information helps us determine that the general form, $$y = a(b)^x$$, must have a **$$b$$-value greater than 1**. --- ### **Summary** - The table confirms **exponential growth** because $$y$$ increases at an increasing rate, and $$b = 1.65$$ (greater than 1). - The equation for the relationship is: $$y = 4(1.65)^x$$ --- ### **Follow-Up Questions** 1. How can you confirm the growth factor $$b$$ using additional points from the table? 2. What would the equation look like if it represented exponential decay instead of growth? 3. Can you predict the value of $$y$$ when $$x = 4$$ using the equation $$y = 4(1.65)^x$$? 4. How does the rate of change differ between linear growth and exponential growth? 5. What are some real-world examples of exponential growth that follow this model? --- ### **Tip** When analyzing exponential functions, always check the initial value ($$a$$) and the growth/decay factor ($$b$$) to determine the behavior (growth or decay) of the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Exponential Growth
Growth Factor
Formulas
y = a(b)^x
Theorems
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Suitable Grade Level
Grades 8-10