Tag Archives: linear function

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

Posted on March 18, 2011 by Guillermo Bautista | 2 Comments

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. Continue reading →

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Posted in High School Mathematics

Tagged 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

Matchsticks, Linear Relations, and Multiple Representations

Posted on October 18, 2010 by Guillermo Bautista | 1 Comment

Introduction

We have mentioned the different types of functions in the Introductions to Functions post. In this post, we are going to learn about linear function and its characteristics.

To start, let us examine the problem below taken from the TIMSS 2003 released items given to Grade 10 students in more than 40 countries all over the world.

Matchsticks are arranged as shown in the figures.

If the pattern is continued, how many matchsticks would be used to make figure 10.

A. 30 B. 33 C. 36 D. 39 E. 42

The problem is too easy that even a first grade pupil would be able to answer it given enough time. Smart students would be able to easily see patterns. For example, they can relate the number of squares to the number of matchsticks. If they cannot find a pattern, the last resort would be by brute force; that is, by manually drawing the tenth figure. Continue reading →

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Posted in High School Algebra, High School Mathematics

Tagged cartesian plane, getting the x-intercept of a function, getting the y-intercept of a function, introduction to function, linear function, linear function definition, linear function example, linear function graph, linear relations, matchsticks and mathematics, math graphs, pattern searching in math, plotting points, representation of functions, TIMSS 2003

Basic Concepts of Functions

Posted on September 29, 2010 by Guillermo Bautista | 2 Comments

Note: This is the second part of the Functions Series. To view the other parts, click the link below.

Part I: Introduction to Functions
Part II: Basic Concepts of Functions

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In the first part of this series, we have discussed that a function is a relationship between two sets where for each value in the first set, there is exactly one corresponding value in the second set. We have painted large cubes, cut them into unit cubes and found a pattern about the number of cubes with 3, 2, 1 and no painted faces.

Figure 1 - Cubes painted and sliced into unit cubes.

We found out that if a cube has side n units, if we painted all of its faces and cut it into unit cubes, the following relationships hold: Continue reading →