linear function Archives - Math and Multimedia

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

November 30, 2015 GB High School Algebra, High School Mathematics

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

Leave a comment linear function, y intercept, y intercept of a linear function

Increasing and Decreasing a in the linear function y = ax

November 19, 2015 GB High School Algebra, High School Mathematics

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

Leave a comment linear function, slope of a function

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

March 18, 2011 GB High School Mathematics

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

2 comments arithmetic sequence, arithmetic sequence equation, arithmetic sequence problems, equation of a line, graphing linear equation, linear equation, linear equation definition, linear equation example, linear function, linear function definition, linear function example, linear function graph, y = ax + b, y = mx + b

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