Graphs of Direct Variation

Sketch $y = 3x$ and $y = 3x + 2$ on the same axes, roughly. They never meet and never cross. Now pick any $x$ and compare $\tfrac{y}{x}$ for the two lines. For one of them the answer is the same wherever you look. Which, and why?

The graph of $y = kx$ is a straight line through the origin, and its gradient is the constant of variation. Both halves of that sentence are needed: a line missing the origin is not direct variation however straight it is.

Substitute $x = 0$ into the equation of direct variation:

$$y = k(0) = 0$$

So the point $(0,0)$ is on the graph, whatever $k$ is. Every direct variation graph passes through the origin, with no exceptions.

That is not a coincidence but a restatement of the definition. Direct variation means the ratio $\tfrac{y}{x}$ is fixed, which forces $y$ to be zero exactly when $x$ is. In context it is usually obvious once said: zero coffees cost zero dollars, zero hours of driving covers zero kilometres, a wire of zero length has zero mass.

The converse matters just as much. If a straight-line graph does not pass through the origin, its equation is $y = mx + c$ with $c \neq 0$, and

$$\frac{y}{x} = \frac{mx + c}{x} = m + \frac{c}{x}$$

which changes as $x$ changes. So the ratio is not fixed and the relationship is not direct variation.

Straight and through the origin. Checking only the first half is the most common error in this lesson.

Take any two points on the line $y = kx$, say $(x_1, kx_1)$ and $(x_2, kx_2)$. The gradient is

$$\frac{\text{rise}}{\text{run}} = \frac{kx_2 - kx_1}{x_2 - x_1} = \frac{k(x_2-x_1)}{x_2-x_1} = k$$

So the gradient of the line is the constant of variation. They are the same number wearing two names, and reading one off a graph gives you the other.

Because the line passes through the origin, there is a shortcut. Take the origin as one of the two points:

$$k = \frac{y - 0}{x - 0} = \frac{y}{x}$$

so the constant is just $\tfrac{y}{x}$ read at any single convenient point on the line. Choose a point where the line crosses a clear grid intersection, and the arithmetic is immediate.

This also settles what a change of $k$ looks like. Larger $k$ means steeper, and a negative $k$ means the line falls from left to right, still through the origin.

Graphing $y = kx$ needs only two points, and one of them is free.

1. Mark the origin, which is always on the line.
2. Choose one convenient value of $x$, work out $y$, and mark that point.
3. Draw the straight line through both, extending it in both directions if negative values make sense.

To graph $y = 2.5x$: the origin, then $x = 4$ gives $y = 10$, so mark $(4,10)$ and join.

Choose the second point to make the arithmetic clean and the point far from the origin. A point close to the origin makes the line's direction very sensitive to a small drawing error; a point near the edge of the axes pins it down much more firmly.

A third point is worth plotting as a check, since three points that fail to line up reveal an arithmetic slip that two points never could.

In a context, restrict the line to the values that make sense. If $x$ is a number of items, the graph is really a set of dots at the whole numbers, and if $x$ is a length then only $x \geq 0$ is meaningful, so the line is drawn from the origin in one direction only.

Given a graph, three questions answer themselves in order.

Is it direct variation? The graph must be a straight line and must pass through the origin. Both.

What is $k$? Read a convenient point $(x, y)$ off the line and compute $\tfrac{y}{x}$.

What is the equation? Write $y = kx$ with the value found.

For a line through the origin passing through $(6, 15)$: it is direct variation, $k = \tfrac{15}{6} = 2.5$, and the equation is $y = 2.5x$.

Comparing two graphs on the same axes is then a comparison of constants. If one line is steeper, its constant is larger, which in context means a higher rate: a more expensive item, a faster speed, a denser material.

Two direct-variation lines on the same axes can only ever cross at the origin. They both pass through it, and two distinct straight lines meet at most once, so if they meet anywhere else they would have to be the same line.

Five items on graphs of direct variation. Work each one, then reveal the answer.

1. 1 A line through the origin passes through $(4, 12)$. Find $k$.

   Show answer
   Gradient from the origin is $\tfrac{y}{x}$.$k = 3$

2. 2 Does a line through $(0,5)$ and $(2,9)$ show direct variation?

   Show answer
   Check where it crosses the vertical axis.No

3. 3 Which is steeper, $y = 4x$ or $y = 0.4x$?

   Show answer
   The gradient is the constant of variation.$y = 4x$

4. 4 Describe the graph of $y = -2x$.

   Show answer
   Negative constant, still through the origin.A line through the origin falling to the right

5. 5 Two direct-variation lines are drawn on one set of axes. Where can they cross?

   Show answer
   Both must pass through one particular point.Only at the origin