Some observations:. Linear relationships are similar to direct variation relationships since the rate of change slope is constant in both, but unlike direct variation relationships, linear relationships can include point 0,k where k is different from 0. Given a relationship between two variables x and y , the relationship is linear if the rate of change is constant, i. If you begin with 15 cookies and one cookie costs 20 cents, the table above shows the relationship between x , the amount you spent, and y , the number of cookies you have.
Use the above application to find runs and rises consistent with slopes of 3, 1, 0. At this time, you should be able to understand the following relationships associated with slopes. When the slope is a fraction, it is not always convenient to think of the run as being equal to 1.
If two variables are directly proportional to one another it means that, as one doubles in size, then so does the other; if one trebles, then so does the other, or if one halves, then so does… well, you get the idea. We say that the proportional change in one variable is equal to the proportional change in the other. In this example, the statement that the acceleration is proportional to the resultant force can be written mathematically as:. Take a look at the data above. Does a double as F doubles?
In the second pair of readings, when F has doubled to 2 N, a has also doubled to 0. Why not? To turn any proportionality relation into an equation, we have to add a constant of proportionality. In maths, this constant is normally given the symbol k , which leads us to the following equation which relates a to F :.
This is done by dividing each value of a by the corresponding value of F. We explain how your work is scored. MAC Introduction and Syllabus. Goals of this Section This is the introduction to the overall course and it contains the syllabus as well as grade information. Goals of this Course This course has several concurrent but different goals. Expect Differences What makes this course different from previous courses?
Methods to Prepare Suggestions on Studying and Learning. General Syllabus This is the syllabus for the course with everything but grading and the calendar.
The Point of Grades This is the grading rubric for the course, including the assignments, how many points things are worth, and how many points are needed for each letter grade.
Mathematical Modeling. Goals of this Section This section is on learning to use mathematics to model real-life situations. Terminology To Know These are important terms and notations for this section. What is mathematical reasoning?
This section aims to introduce the idea of mathematical reasoning and give an example of how it is used. Logical Deduction This section analyzes the previous example in detail to develop a three phase deductive process to develop a mathematical model.
Types of Information This section aims to explore and explain different types of information. Is this actually math? Math as a Language This section contains important points about the analogy of mathematics as a language.
Numeric Model Walkthrough This is a detailed numeric model example and walkthrough. Embrace Laziness! Variables and Their Roles This section explains types and interactions between variables. Generalized Model Walkthrough This is an example of a detailed generalized model walkthrough. Variables, Functions, Graphing, and Universal Properties. Goals of this Section This section is on functions, their roles, their graphs, and we introduce the Library of Functions.
Relationship vs. Equations In this section we discuss a very subtle but profoundly important difference between a relationship between information, and an equation with information. Functions In this section we discuss what makes a relation into a function. Functions Require Context In this section we demonstrate that a relation requires context to be considered a function.
Set Notation In this section we cover how to actual write sets and specifically domains, codomains, and ranges. Function Notation This section covers function notation, why and how it is written. Graphing Introduction This section introduces graphing and gives an example of how we intuitively use it. French History and Dinosaurs! This section introduces the origin an application of graphing. Graphing To Relate Variables This section describes how we will use graphing in this course; as a tool to visually depict a relation between variables.
Using Graphs This section covers what graphs should be used for, despite being imprecise. Vertical Line Test This section describes the vertical line test and why it works. Parent Functions This section provides the specific parent functions you should know. In other words, for every value of x , there is only one corresponding value of y.
When you assign a value to the independent variable, x , you can compute the value of the dependent variable, y. You can then plot the points named by each x , y pair on a coordinate grid. Students should already know that any two points determine a line. So graphing a linear equation in fact only requires finding two pairs of values and drawing a line through the points they describe.
All other points on the line will provide values for x and y that satisfy the equation. The graphs of linear equations are always lines. However, it is important to remember that not every point on the line that the equation describes will necessarily be a solution to the problem that the equation describes.
For example, the problem may not make sense for negative numbers say, if the independent variable is time or very large numbers say, numbers over if the dependent variable is grade in class. In this equation, for any given steady rate, the relationship between distance and time will be linear.
However, distance is usually expressed as a positive number, so most graphs of this relationship will only show points in the first quadrant. Notice that the direction of the line in the graph below is from bottom left to top right. Lines that tend in this direction have positive slope.
A positive slope indicates that the values on both axes are increasing from left to right. In this equation, since you won't ever have a negative amount of water in the bucket, the graph will show points only in the first quadrant.
Notice that the direction of the line in this graph is top left to bottom right. Lines that tend in this direction have negative slope. A negative slope indicates that the values on the y- axis are decreasing as the values on the x- axis are increasing.
Again in this graph, we are relating values that only make sense if they are positive, so we show points only in the first quadrant. Moreover, in this case, since no polygon has fewer than 3 sides or angles and the number of sides or angles of a polygon must be a whole number, we show the graph starting at 3,3 and indicate with a dashed line that points between those plotted are not relevant to the problem. Since it's perfectly reasonable to have both positive and negative temperatures, we plot the points on this graph on the full coordinate grid.
The slope of a line tells two things: how steep the line is with respect to the y- axis and whether the line slopes up or down when you look at it from left to right.
More technically speaking, slope tells you the rate at which the dependent variable is changing with respect to the change in the independent variable. Pick any two points on the line. To find how fast y is changing, subtract the y value of the first point from the y value of the second point: y 2 — y 1. To find how fast x is changing, subtract the x value of the first point from the x value of the second point: x 2 — x 1.
It does not matter which points along the line you designate as A and B , just as long as we're consistent with which is the "first" point x 1 , y 1 and which is the "second" x 2 , y 2. It is also the same value you will get if you choose any other pair of points on the line to compute slope. The equation of a line can be written in a form that makes the slope obvious and allows you to draw the line without any computation.
If students are comfortable with solving a simple two-step linear equation, they can write linear equations in slope-intercept form. In the equation, x and y are the variables. The numbers m and b give the slope of the line m and the value of y when x is 0 b. The value of y when x is 0 is called the y -intercept because 0, y is the point at which the line crosses the y -axis. You can draw the line for an equation matching this linear formula by plotting 0, b , then using m to find another point.
Now look at b in the equation: —3 should be where the line intercepts the y -axis, and it is. When a line slopes up from left to right, it has a positive slope. This means that a positive change in y is associated with a positive change in x. The steeper the slope, the greater the rate of change in y in relation to the change in x. When the line represents real-world data points plotted on a coordinate plane, a positive slope indicates a positive correlation, and the steeper the slope, the stronger the positive correlation.
Consider a linear equation where the independent variable g is gallons of gas used and the dependent variable d is the distance traveled in miles.
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