en Vision Algebra 2 Name PearsonRealize.com 4-1 Additional Practice Inverse Variation and the Reciprocal Function Do the tables below represent a direct variation or an inverse variation?
The correct answer and explanation is:
To determine whether the tables represent a direct variation or an inverse variation, let’s first define both terms:
Definitions:
- Direct Variation: A relationship where two variables xx and yy are proportional, meaning y=kxy = kx, where kk is a constant. In this case, as xx increases, yy increases (or both decrease simultaneously), maintaining the constant ratio yx=k\frac{y}{x} = k.
- Inverse Variation: A relationship where the product of two variables xx and yy is constant, meaning xy=kxy = k or y=kxy = \frac{k}{x}. Here, as xx increases, yy decreases, and vice versa, maintaining the constant product xy=kxy = k.
Step-by-step Analysis:
To analyze the tables:
- Check if the ratio yx\frac{y}{x} remains constant (direct variation).
- Check if the product xyxy remains constant (inverse variation).
Example Table:
| xx | yy |
|---|---|
| 1 | 12 |
| 2 | 6 |
| 3 | 4 |
| 4 | 3 |
Testing for Direct Variation:
- Compute yx\frac{y}{x}: 121=12,62=3,43≠constant\frac{12}{1} = 12, \quad \frac{6}{2} = 3, \quad \frac{4}{3} \neq \text{constant}
- The ratio yx\frac{y}{x} is not constant, so not a direct variation.
Testing for Inverse Variation:
- Compute xyxy: 1⋅12=12,2⋅6=12,3⋅4=12,4⋅3=121 \cdot 12 = 12, \quad 2 \cdot 6 = 12, \quad 3 \cdot 4 = 12, \quad 4 \cdot 3 = 12
- The product xyxy is constant, so this is an inverse variation.
Explanation :
A table represents direct variation if the ratio yx\frac{y}{x} is constant, meaning yy increases or decreases proportionally with xx. For example, if x=1,2,3x = 1, 2, 3 and y=2,4,6y = 2, 4, 6, the ratio yx=2\frac{y}{x} = 2 remains constant, indicating direct variation.
In contrast, a table represents inverse variation if the product xyxy is constant. This relationship implies that as xx increases, yy decreases, and vice versa. For instance, if x=1,2,4x = 1, 2, 4 and y=8,4,2y = 8, 4, 2, the product xy=8xy = 8 remains the same.
To determine the type of variation, analyze the given data. Calculate yx\frac{y}{x} for all pairs in the table. If constant, it is a direct variation. Otherwise, compute xyxy; if constant, it is an inverse variation. If neither is constant, the table does not represent either type of variation.
For example, in the table where x=1,2,3,4x = 1, 2, 3, 4 and y=12,6,4,3y = 12, 6, 4, 3, the ratio yx\frac{y}{x} varies (12, 3, etc.), eliminating direct variation. However, the product xy=12xy = 12 for all entries, confirming inverse variation.
Understanding these patterns helps identify relationships between variables, providing insights into their mathematical behavior.