The Science of Life Lecture Series

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  This entry is an introduction to Linear Functions in both graphs and models. There are two situations where the slope would not be stated in the slope-intercept form.  These are cases where the line crosses only one of the 2 (or more) axes. First, let's consider the horizontal line.  If we have a line which has a y-intercept but does not have an x-intercept (does not cross the x-axis), it is called a horizontal line.  Let's look at it from the point of view of the slope-intercept form we developed last time, $y=mx+b$.  We know that the slope is the change in the y-value per change in the x-value.  If we have two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, the slope is mathematically defined as $m=\frac{y2-y1}{x2-x1}$.  The denominator, the $ dx=x2-x1$, is never going to be zero when there's a y-intercept, since we can go from the left of the y-intercept to th...

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  This entry is an introduction to Linear Functions in both graphs and models. Linear Equations are the easiest functions to conceptualize. They are lines which stretch out to infinity in both directions. On an x-y plane, the equation of a line can follow many forms, the first one usually introduced being the slope-intercept form y=mx+b, where m is the slope (change in the y-coordinate per 1 unit change in the x-coordinate) and b is the y-intercept (point where it crosses the y-axis). There are other forms of the equation of the line, but I'll get to those in future entries. If we have multiple points which form an obvious line, we can use two of those points to find the slope of the line. As mentioned earlier, the slope is the motion along the y-axis as the x-coordinate changes. With that in mind, we can use two points (x1, y1) and (x2, y2) to determine the slope. The slope is given by...

Hello internet, and welcome to the Algebra Lecture Series from The Science of Life.  In this entry, we move to the definition of an algebraic function. I feel it is a good idea to start with the definitions of domain and range.  The domain is defined as the set of all possible values for which the independent variable (the input, the x-value) can take up.  This is all possible values for the horizontal (x) axis.  The range is the set of all possible values for which the dependent variable (the output, the y-value) can take up.  This is all possible values for the vertical (y) axis. The blue horizontal axis (typically the x-axis) is the domain and the independent variable. The green vertical axis (typically the y-axis) is the range and the dependent variable. Now we move to the concept of a function.  A good definition of the function is any equation where each input value has exactly one output value.  The technical version of this is "any eq...

If you're familiar with a number line, then you know that each point on that number line corresponds to a number.  If we take two number lines which are perpindicular to one another (they cross each other to make a right angle) and have them cross at the others zero-point, then we have a method of analyzing two-dimensional data-sets and equations, data and equations with two variables which relate to one another.  This two-number-line situation is typically called either the x-y coordinate system (because the horizontal number line is labeled as the x-number line and the horizontal is labeled as the y-number-line) or the Cartessian Coordinate System (named after French mathematician René Descartes, supposedly the first person to come up with the system).  The two number lines, since they're in the context of a coordinate system, are each now called an axis (plural: axes). The x-axis and y-axis of a Cartesian coordinate system. Any point on this coordinate system i...

Graphs are an important part of Algebra since they allow us to visualize information from the equations of Algebra.  After all, we deal better with the visual than the equation.  When we have an equation of two variables, we can see how they relate to one another more readily in a graph than in an equation. The course of Algebra focuses on the concept of functions (which will be defined in Section 2; because yes, functions are different than equations).  Most of the graphs considered here are what is called linear equations, equations which produce lines.  This chapter has the purpose of not only teaching the basics of graphing functions, but also using them for problem solving. Section 1: Graphs Section 2: Functions Section 3: Linear Functions: Graphs and Models Section 4: Other Equations of Line Section 5: Other Equations of Lines Section 6: The Algebra of Functions That's the end of this section.  If you have any questions, please leave ...

In all of science, there are prefixes which are used to are attached to units in order to have a word representation of having differing multiples of 10.  An example of this is the fact that a kilometer is one meter times 1000, or one meter times "three multiples of 10" ($10^3$).  We can also go the other way; a millimeter is one meter divided by 1000, or one meter divided by "three multiples of 10" ($10^3$).  When we have a situation of "divided by $10^3$", as mentioned in the Algebra Lecture Series, it is equivalent to saying "multiplied by $10^-3$".  Notice that the "divided by" has a positive exponent while the "divided by" has a negative exponent.  That stems from the exponent subtraction rule in Algebra. Here's the list of all of the prefixes and what they represent in two numeric forms: Prefix Symbol Multiplier Exponential yotta Y 1,000,000,000,000,000,000,000,0...

One thing to know about Chemistry -- and all of science -- is that unless one group of scientists are intentionally trying to experimentally confirm the results of a specific experiment done by another group of scientists, no two scientists typically approach the same question the same way.  Even as this may be the case, all scientists are using the same general scientific method.  The scientific method is the backbone of all scientific experiments and gathering of all scientific knowledge.  In very much the same way all humans look pretty similar (but not exactly the same), the scientific method makes it so the experiments that two different scientists perform on the same concept with look similar, but not exactly the same. When we run an experiment, we start by collecting information.  How do things change if I slowly change this parameter only?  How about if I change that parameter only?  Now that I have the previous two questions answered, now let...