Question

which graph matches this direct variation equation?
$y = -\frac{3}{4}x$

Explanation:

Step1: Recall direct variation graph properties

Direct variation equations are of the form \( y = kx \), where \( k \) is the constant of variation. The graph is a straight line passing through the origin \((0,0)\) because when \( x = 0 \), \( y = 0 \). Also, the slope \( k \) determines the direction and steepness. Here, \( k=-\frac{3}{4}\), which is negative, so the line should have a negative slope (decreasing from left to right) and pass through \((0,0)\).

Step2: Analyze each graph

• First graph: Does it pass through \((0,0)\)? No, it has a y - intercept (crosses the y - axis at a non - zero point). So this is not the graph of \( y =-\frac{3}{4}x\).
• Second graph: The slope is positive (the line is increasing from left to right), but our slope \( k =-\frac{3}{4}\) is negative. So this is not the correct graph.
• Third graph: It passes through the origin \((0,0)\) (since when \( x = 0\), \( y = 0\)) and has a negative slope (decreasing from left to right), which matches the equation \( y=-\frac{3}{4}x\) (where the slope \( m =-\frac{3}{4}\) is negative and the line passes through the origin).

Answer:

The third graph (the one with the line passing through the origin and having a negative slope, decreasing from left to right)