The effect of the sign of the slope in y = ax

A linear function is a function whose equation is of the form y = ax + b. We separate the discussion about it into two parts: b = 0 and b \neq 0. In this post, we only discuss the graph of y = ax + b where b = 0. We discuss the effect of the sign of the slope in y = ax.

If we let b = 0, the equation y = ax + b becomes y = ax + 0 or simply y = ax.

Notice that if x = 0, then y = ax = a(0) = 0. This means that the graph contains the point with coordinates (0,0). Therefore, y = ax passes through the origin.

Generalization 1: The graph y = ax passes through the origin.

We now examine the effect of the values of a. There are three cases: a = 0, a > 0, and a < 0

Case 1: a = 0

If a = 0, then y = 0(x) = 0. This means that for all values of x, the value of y = 0. This gives us the horizontal line y = 0.

Case 2: a > 0

If we let x > 0, then ax > 0 since a > 0. That is, we multiplied a positive number a by a positive number x which means that their product is positive. In effect, if x > 0, then y >0. This only means that the graph passes through the first quadrant.

If we let x < 0, then ax > 0 since a > 0. That is, we multiplied a positive number a by a negative number x which means that their product is negative. In effect, if x < 0, then y < 0. This only means that the graph passes through the third quadrant.

So, the graph y = ax where a > 0 passes through the origin (Generalization 1) and parts of it are on the first and third quadrant.

sign of the slope

Generalization 2: If a > 0, the graph y = ax passes through the origin, the first quadrant, and the third quadrant.

Case 3: a < 0

If we let x > 0, then ax < 0 since a < 0. That is, we multiplied a negative number a by a positive number x which means that their product is negative. In effect, if x > 0, then y < 0. This only means that the graph passes through the fourth quadrant.

If we let x < 0, then ax > 0 since a < 0. That is, we multiplied a negative number a by a negative number x which means that their product is positive. In effect, if x < 0, then y > 0. This only means the graph passes through the second quadrant.

So, the graph y = ax where a < 0 passes through the origin (Generalization 1) and parts of it are on the second and fourth quadrant.

sign of the slope

Generalization 3: If a < 0, the graph of y = ax 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.

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