Graph Direct Variation Equations - Algebra-2

1. Fundamental Concepts

Definition: The graph of a direct variation equation is a graph formed by all points in the coordinate plane that satisfy the direct variation relationship (where k is a non-zero constant of variation).
Essence: It is a visual representation of the direct proportional relationship between two variables, and it is a special form of linear function graphs.

2. Key Concepts

Shape of the Graph: The graph of the direct variation function is a straight line passing through the origin . This is because when , according to the relationship, so the origin is a fixed point on the graph.
Relationship with Slope: The constant of variation k is also the slope of the straight line, which determines the steepness and direction of the line:
- When , the line slopes upward from left to right (y increases as x increases) and passes through the first and third quadrants;
- When , the line slopes downward from left to right (y decreases as x increases) and passes through the second and fourth quadrants.
Difference from Non-Direct Variation Graphs: A straight line that does not pass through the origin (e.g., with ) is definitely not the graph of a direct variation function.

3. Examples

Easy Level

Judge: Is the graph of the function a graph of a direct variation function? Please explain the reason.

Answer: Yes.

Explanation: The function conforms to the form of a direct variation function (where ), and its graph is a straight line passing through the origin, so it is a graph of a direct variation function.

Medium Level

It is known that a straight line passes through the points and . Judge whether the straight line is the graph of a direct variation function. If it is, find its constant of variation.

Answer: Yes; the constant of variation is .

Explanation: The straight line passes through the origin, and we can set its corresponding function relationship as . Substituting the point into it, we get , that is, , which conforms to the form of the direct variation function . Therefore, it is the graph of a direct variation function, and the constant of variation is .

Hard Level

There are two straight lines: the function relationship of line A is , and line B passes through the points and . Judge which straight line is the graph of a direct variation function and explain the reason.

Answer: Line B is the graph of a direct variation function.

Explanation: The relationship of line A is , which contains the constant term 1 and does not conform to the form . Moreover, its graph does not pass through the origin, so it is not the graph of a direct variation function; The two points passed by line B satisfy and , that is, the constant of variation , and the corresponding function relationship is , which conforms to the form of a direct variation function, and its graph passes through the origin. Therefore, line B is the graph of a direct variation function.

4. Problem-Solving Techniques

Methods to Judge Whether a Graph is That of a Direct Variation Function:
- Observe whether the graph is a straight line;
- Check whether the straight line passes through the origin ;
- Verify whether the ratio of any point (except the origin) on the graph is the same non-zero constant (i.e., whether there is a fixed constant of variation k).
Steps to Draw the Graph of a Direct Variation Function:
- Determine two points: the origin is a mandatory point; select a non-zero x-value (e.g., ), substitute it into to find the corresponding y-value, and obtain the second point ;
- Use a ruler to connect these two points and extend them infinitely in both directions to get the graph of the direct variation function.
Method to Find the Constant of Variation k from the Graph:
- Select any point (with ) on the graph except the origin, and calculate according to to get the constant of variation.