Match each statement with its corresponding graph in choices A–D. In each case, k \u003e 0. y varies directly as x. (y=kx)

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. Recognize the shape of the graph for $$y = kx$$ with $$k \u003e 0$$: this is a straight line passing through the origin (0,0) with a positive slope $$k. $$Identify the key features of the graph: since $$k \u003e 0$$, the line rises from left to right, meaning as $$x$$ increases, $$y$$ also increases proportionally. 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. Match the statement "y varies directly as x" to the graph that fits these criteria, confirming it represents the equation $$y = kx$$ with $$k \u003e 0.$$

Ch. 3 - Polynomial and Rational Functions

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 17

Chapter 4, Problem 17

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

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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.

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 $k$.

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.

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.

Match the statement "y varies directly as x" to the graph that fits these criteria, confirming it represents the equation $y = kx$ with $k > 0$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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