Question

which graph matches this direct variation equation?
y = -3x

Explanation:

Step1: Recall direct variation graph properties

A direct variation equation \( y = kx \) (here \( k = -3 \)) is a straight line passing through the origin \((0,0)\) (since when \( x = 0 \), \( y = 0 \)). The slope \( k \) determines the direction and steepness. A negative slope (\( k=-3<0 \)) means the line falls from left to right.

Step2: Analyze slope magnitude and direction

The slope \( |k| = 3 \), which is a relatively steep slope (steeper than a slope with \( |k| = 1 \), for example). Let's check the three graphs:

• First graph: Slope is positive (rises from left to right), so it can't be \( y=-3x \) (needs negative slope).
• Second graph: Passes through origin, has negative slope, and steep (since for \( x = 1 \), \( y=-3(1)=-3 \); for \( x=-1 \), \( y=-3(-1)=3 \), which matches the points on this graph).
• Third graph: Slope is negative but very shallow (closer to slope of \(-\frac{1}{3}\) or similar), not steep like \( -3 \).

Answer:

The second graph (middle one) matches \( y = -3x \).