A **linear function** is a function whose equation is of the form . We separate the discussion about it into two parts: and . In this post, we only discuss the graph of where . We discuss the effect of the sign of the slope in .

If we let , the equation becomes or simply .

Notice that if , then . This means that the graph contains the point with coordinates . Therefore, passes through the origin.

*Generalization 1*: *The graph***passes through the origin**.

We now examine the effect of the values of . There are three cases: , , and .

**Case 1:**

If , then . This means that for all values of , the value of . This gives us the horizontal line .

**Case 2:**

If we let , then since . That is, we multiplied a positive number by a positive number which means that their product is positive. In effect, if , then . This only means that the graph passes through the first quadrant.

If we let , then since . That is, we multiplied a positive number by a negative number which means that their product is negative. In effect, if , then . This only means that the graph passes through the third quadrant.

So, the graph where passes through the origin (Generalization 1) and parts of it are on the first and third quadrant.

** Generalization 2:** If ,

**the graph**

**passes through the origin, the first quadrant, and the third quadrant.****Case 3:**

If we let , then since . That is, we multiplied a negative number by a positive number which means that their product is negative. In effect, if , then . This only means that the graph passes through the fourth quadrant.

If we let , then since . That is, we multiplied a negative number by a negative number which means that their product is positive. In effect, if , then . This only means the graph passes through the second quadrant.

So, the graph where passes through the origin (Generalization 1) and parts of it are on the second and fourth quadrant.

* Generalization 3:* If , the graph of passes through the origin, the first quadrant, and the third quadrant.

In the next post, we are going to relate the increase and decrease of a in the graph of the function y = ax.