Math Problem Statement
A table showing pairs of x and y values is shown below.
x y 0 0.5 1 1.5 2 4.5 3 13.5 4 40.5 Which equation describes the relationship between the pairs of x and y values? A. y = (0.5-3) B. y = 0.5.3 C. y=2x0.5 D. y = x +0.5
Solution
Let's carefully analyze the given values and identify the relationship between and .
Table of values:
x & : 0, 1, 2, 3, 4 \\ y & : 0.5, 1.5, 4.5, 13.5, 40.5 \end{aligned}$$ ### Identifying the pattern: - Notice how the $$y$$-values seem to increase rapidly, hinting at a potential **exponential** or **multiplicative relationship**. - Let's examine the ratio between consecutive $$y$$-values: $$\frac{1.5}{0.5} = 3, \quad \frac{4.5}{1.5} = 3, \quad \frac{13.5}{4.5} = 3, \quad \frac{40.5}{13.5} = 3$$ - The $$y$$-values are multiplied by **3** each time $$x$$ increases by 1. This suggests the relationship follows the **exponential pattern**: $$y = 0.5 \cdot 3^x$$ ### Checking the answer options: - **A. $$y = (0.5 - 3)$$**: Incorrect, this is not a valid equation. - **B. $$y = 0.5 \cdot 3$$**: Incorrect, it is a constant value rather than a function of $$x$$. - **C. $$y = 2x0.5$$**: Incorrect, it’s not exponential and doesn't fit the data. - **D. $$y = 0.5 \cdot 3^x$$**: This equation matches our findings perfectly! Thus, the correct answer is: $$\boxed{D. \, y = 0.5 \cdot 3^x}$$ --- Would you like more details or have any questions? Here are 5 related questions to deepen your understanding: 1. What is the general form of an exponential function? 2. How do you determine if a relationship is linear or exponential from a table of values? 3. What is the difference between an arithmetic and a geometric sequence? 4. How would the equation change if the initial $$y$$-value were 1 instead of 0.5? 5. Can exponential functions have negative values for $$y$$? Why or why not? **Tip:** When the ratio between consecutive outputs is constant, the function is likely exponential.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential Functions
Geometric Sequences
Formulas
y = 0.5 * 3^x
Theorems
Exponential Growth Theorem
Geometric Progression
Suitable Grade Level
Grades 9-12