Question

find the graph for this direct variation equation. $y = \frac{2}{5}x$

Explanation:

Step1: Recall Direct Variation Graph Properties

A direct variation equation \( y = kx \) (where \( k \) is the constant of variation) is a linear equation with a slope of \( k \) and a y - intercept of \( 0 \) (passes through the origin \((0,0)\)). For the equation \( y=\frac{2}{5}x \), the slope \( k = \frac{2}{5}\) (positive, so the line should rise from left to right) and it passes through \((0,0)\).

Step2: Analyze the Slope

The slope \( \frac{2}{5}\) means that for every \( 5\) units we move to the right (increase in \( x\) by \( 5\)), we move up \( 2\) units (increase in \( y\) by \( 2\)). Or for \( x = 5\), \( y=\frac{2}{5}\times5 = 2\). Let's check the graphs:

• The third graph has a negative slope (line falls from left to right), so it can be eliminated.
• For the first graph, let's check the rate of rise. If we take \( x = 5\), the \( y\) - value in the first graph seems to be more than \( 2\) (since the line is steeper).
• For the middle graph, when \( x = 5\), \( y=\frac{2}{5}\times5=2\), which matches the slope of \( \frac{2}{5}\) (the line rises gradually from the origin and reaches \( y = 2\) when \( x = 5\)).

Answer:

The Middle Graph