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.

adding b to y = ax + b

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

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

Equation of a line: The derivation of y = mx + b

We have discussed in context the origin (click here and here) of the linear equation y = ax + b, where a and b are real numbers.  We have also talked about the slope of a line and many of its properties. In this post, we will discuss the generalization of the equation of a line in the coordinate plane based on its slope and y-intercept.

We have learned that to get a slope of a line, we only need two points.  We have also learned that given two points on a line, its slope is described as the rise (difference in the y-coordinates) over the run (difference in x-coordinates).  Therefore, if we have two points with coordinates (x_1,y_1) and (x_2,y_2), the slope m is  defined the formula

m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}.

All the points on a vertical line have similar x-coordinates; therefore, the run {x_2 - x_1} is equal to 0 making m undefined.  From here, we can conclude a vertical line has no slope. » Read more

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