Math Problem Statement
The table below shows which type of variation? The table contains values of x (1, 2, 3, 4, 5) and corresponding values of y (-4, -2, -4/3, -1, -4/5).
Solution
The table in the question appears to represent pairs of values that need to be analyzed for a type of variation. Variations can include direct variation, inverse variation, or neither.
Let’s analyze the data from the table step by step:
Table:
x & y \\ \hline 1 & -4 \\ 2 & -2 \\ 3 & -\frac{4}{3} \\ 4 & -1 \\ 5 & -\frac{4}{5} \\ \end{array}$$ ### Step 1: Check for Direct Variation Direct variation means $$y = kx$$, where $$k$$ is a constant. Let’s calculate $$\frac{y}{x}$$ for each pair: - For $$x = 1, y = -4$$: $$\frac{y}{x} = \frac{-4}{1} = -4$$ - For $$x = 2, y = -2$$: $$\frac{y}{x} = \frac{-2}{2} = -1$$ - For $$x = 3, y = -\frac{4}{3}$$: $$\frac{y}{x} = \frac{-\frac{4}{3}}{3} = -\frac{4}{9}$$ - For $$x = 4, y = -1$$: $$\frac{y}{x} = \frac{-1}{4} = -\frac{1}{4}$$ - For $$x = 5, y = -\frac{4}{5}$$: $$\frac{y}{x} = \frac{-\frac{4}{5}}{5} = -\frac{4}{25}$$ Since $$\frac{y}{x}$$ is not constant, this is **not a direct variation**. --- ### Step 2: Check for Inverse Variation Inverse variation means $$y = \frac{k}{x}$$, where $$k$$ is a constant. Let’s calculate $$x \cdot y$$ for each pair: - For $$x = 1, y = -4$$: $$x \cdot y = 1 \cdot (-4) = -4$$ - For $$x = 2, y = -2$$: $$x \cdot y = 2 \cdot (-2) = -4$$ - For $$x = 3, y = -\frac{4}{3}$$: $$x \cdot y = 3 \cdot (-\frac{4}{3}) = -4$$ - For $$x = 4, y = -1$$: $$x \cdot y = 4 \cdot (-1) = -4$$ - For $$x = 5, y = -\frac{4}{5}$$: $$x \cdot y = 5 \cdot (-\frac{4}{5}) = -4$$ Since $$x \cdot y = -4$$ is constant, this is an **inverse variation**. --- ### Conclusion: The table represents **inverse variation**, where $$y = \frac{-4}{x}$$. Would you like a more detailed explanation or visual graph for this? --- ### 5 Related Questions: 1. What is the formula for direct variation, and how does it differ from inverse variation? 2. How do you graph an inverse variation equation? 3. What are some real-life examples of inverse variation relationships? 4. How do you test whether a dataset represents a quadratic relationship instead of a variation? 5. How can you use logarithms to analyze complex variations? ### Tip: When analyzing variation, always test both $$\frac{y}{x}$$ and $$x \cdot y$$ to check for direct and inverse relationships.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Variation
Inverse Variation
Formulas
y = kx
y = k / x
Theorems
Inverse Variation Principle
Suitable Grade Level
Grades 8-10
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