Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as x. (y=kx)

Verified step by step guidance

1. Identify the equation given: \( y = kx \), where \( k > 0 \). This represents a direct variation between \( y \) and \( x \).
2. Understand that in a direct variation, as \( x \) increases, \( y \) also increases proportionally, and the graph is a straight line passing through the origin (0,0).
3. Recognize that the slope of the line is \( k \), which is positive since \( k > 0 \). This means the line will have an upward slope.
4. Look for a graph that is a straight line passing through the origin with a positive slope.
5. Match the description of the graph to one of the choices A–D, ensuring it is a line through the origin with a positive slope.

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where one variable is a constant multiple of the other. In the equation y = kx, 'k' represents the constant of variation, indicating how much y changes for a unit change in x. This relationship implies that as x increases, y also increases proportionally, and vice versa.

Graph of a Linear Function

The graph of a linear function, such as y = kx, is a straight line that passes through the origin (0,0) when k is positive. The slope of the line is equal to the constant 'k', which determines the steepness of the line. A higher value of k results in a steeper line, while a lower value leads to a gentler slope.

Positive Slope

A positive slope indicates that the line rises as it moves from left to right on a graph. In the context of the equation y = kx with k > 0, this means that for every increase in x, y also increases, reflecting a direct and positive relationship between the two variables. This characteristic is essential for identifying the correct graph that represents the direct variation.