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. Understand the concept of direct variation: when we say "y varies directly as x," it means that y is proportional to x, and the relationship can be written as \(y = kx\), where \(k\) is a positive constant.
2. Recognize the shape of the graph for \(y = kx\) with \(k > 0\): this is a straight line passing through the origin (0,0) with a positive slope \(k\).
3. Identify the key features of the graph: since \(k > 0\), the line rises from left to right, meaning as \(x\) increases, \(y\) also increases proportionally.
4. Compare each graph choice (A–D) to these characteristics: look for the graph that is a straight line through the origin with a positive slope.
5. Match the statement "y varies directly as x" to the graph that fits these criteria, confirming it represents the equation \(y = kx\) with \(k > 0\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another, expressed as y = kx with k > 0. This means as x increases, y increases proportionally, resulting in a straight line through the origin.

Slope of a Linear Function

The slope represents the rate of change of y with respect to x in a linear function. For y = kx, the slope is k, which determines the steepness and direction of the line. Since k > 0, the line rises from left to right.

Graphing Linear Equations

Graphing y = kx involves plotting points where y is proportional to x, producing a straight line through the origin (0,0). Identifying the correct graph requires recognizing this linear pattern and positive slope among the options.