Question

Graph the equation. State whether the two quantities have direct variation. If they have direct variation, find the constant of variation and the slope of the direct variation model. $$y=\frac{1}{2} x$$

Short answer

Yes, the given equation \(y = \frac{1}{2} x\) represents a direct variation. The constant of variation and the slope of the direct variation model is \(\frac{1}{2}\). The graph would be a straight line passing through the origin with a slope of \(\frac{1}{2}\).

Step-by-step solution

Identify the direct variation

The given equation is \(y = \frac{1}{2} x\). This matches the form \(y = kx\), hence there is a direct variation.

Identify the constant of variation

The constant of variation \(k\) is the coefficient of \(x\) in the equation. From the equation \(y = \frac{1}{2} x\), the constant of variation is \(\frac{1}{2}\).

Identify the slope of the direct variation model

For a direct variation equation, the slope is same as the constant of variation. So, the slope of the line representing the direct variation is also \(\frac{1}{2}\).

Graph the equation

On the graph, plot the line that passes through the point (0, 0) and has a slope of \(\frac{1}{2}\). This line represents the direct variation model of the given equation.