Increasing and Decreasing a in the linear function y = ax

In the previous post, we have learned the effect of the sign of a in the linear function y = ax. In this post, we learn the effect of increasing and decreasing the value of a. Since we have already learned that if a = 0, the graph is a horizontal line, we will discuss 2 cases in this post: a > 0, and a < 0.

Case 1: a > 0

Let us consider several cases of the graph of y = ax where a > 0. Let the equation of the functions be f(x) = \frac{1}{2}x, g(x) = x, and h(x) = 2x making a = \frac{1}{2}, 1, and 2, respectively. As we can see from the table, for the same x > 0, the larger the slope, the larger its corresponding y-value. This means that for x = 1, the point (1,h(1)) is above (1, g(1)) and that the point (1,g(1)) is above (1,f(1)). We can say that as x increases, h is increasing faster than g, and g is increasing faster than the increase in f» Read more

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» Read more

The Basics of Inverse Proportion

In the previous post, we have discussed the basics of direct proportions. Recall that when two quantities x and y change and if x changes n times, then y changes n times, then we can say that y is directly proportional to x. In this post, we are going to learn about inverse proportions.

Problem

A rectangle has area 24 square units. Find the possible areas if the length and width are both whole numbers.

Solution and Discussion

The table shows the pairs of length and width that has area of 24 square units.  » Read more

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