The effect of adding b to the linear function y = ax + b

In the previous post, we have learned about the effects of a in the linear function with equation y = ax. In this post, we learn about the effects of adding b to that equation. That is, we want to learn the effects of b in the linear function with equation y = ax + b.

Consider the graph of the functions y = x, y = x + 2 and y = x – 3. The table of values (click figure to enlarge) below shows the corresponding y values of the three linear functions. The effect of adding 2 to the function y = x adds 2 to all the y values of y = x. This implies that in the graph, all the points with corresponding x values are moved 2 units above the graph of y = x. In addition, in the graph of y = x – 3, the -3 subtracts 3 from all the y values of y = x. In effect, all the points  with corresponding x values are moved 3 units below the graph of y = x. In addition, for y = x, if x = 0, y = 0. That means that the graph passes through the origin. On the other hand, for y = x + 2, when x = 0, then y = 0 + 2 = 2. This means that the graph passes through y = 2. Further, for y = 0 – 3 = -3. This means that the graph passes through y = -3. These are shown both in the table above and in the graph below. » Read more

The Latex path not specified WordPress bug

If you have observed, some equations that use Latex contain the error “latex path specified error.” This is WordPress but itself and I can’t really do anything about it for now. This is not only happening in my blog but in other people’s blogs as well. I think it has something to do with the Beautiful Math of the Jetpack Plugin of WordPress. For now, we’ll just have to wait it out until WordPress fixes this error.

If you are a math blogger and experiencing the same thing, there are some discussions in the links below.

https://en.forums.wordpress.com/topic/latex-path-not-specified-errors

A common remedy can be done using another plugin, but if you have a lot of posts, I don’t think this is a good idea.

https://wordpress.org/support/topic/latex-path-not-specified-errors

For now, I will be writing about things that require minimal use of latex.

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 1 2 3 4 5 22