Linear Functions: Graphs and Models: The Algebra Lecture Series Chapter 2 Section 3
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.
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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 $m=\frac{y_2-y_1}{x_2-x_1}$. Realistically, you could use either point as the first point, so long as the point for the first y-value is also the coordinate for the first x-value and the point for the second y-value is also the coordinate for the second x-value. So realistically, we could write $m=\frac{y_2-y_1}{x_2-x_1}=\frac{y_1-y_2}{x_1-x_2}$.
The intercepts are the points that cross the axes. The y-intercept is the point where the line crosses the y-axis and the x-intercept is where the line crosses the x-axis. Since the intercept is where the line crosses that coordinate's axis, the other coordinate is 0. That means the x-intercept is at "ground level" of the coordinate system, which means that it is not above ground (more positive than zero) nor below ground (more negative than zero). That means the y-coordinate at the x-axis is 0.
There are two similar ways to graph a line. The first is to choose two arbitrary numbers for x, plug them into the equation to find their corresponding y values, plot the points, and then draw a line that goes to infinity passing through those two points. The second is a specialized version of the first method. This second method is to find the two intercepts of the line, plot those points, and draw the line going to infinity through those two intercept points.
In these kinds of linear models, the slope is often referred to as the rate of change of y with respect to x. This is because the slope is a measure of how much y changes for every unit x changes. This means that this form is useful for modelling. For example, the rate of brain developement issues as a result of lead consumed can be described as 5 failed neuron connections per ppt of lead in the water. (The numbers are not true, but on average, the concept still holds true. Also, this is a simplified model.) So the model would be $y(n)=5 \frac{n}{ppt}*x(ppt)+0$.
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 $m=\frac{y_2-y_1}{x_2-x_1}$. Realistically, you could use either point as the first point, so long as the point for the first y-value is also the coordinate for the first x-value and the point for the second y-value is also the coordinate for the second x-value. So realistically, we could write $m=\frac{y_2-y_1}{x_2-x_1}=\frac{y_1-y_2}{x_1-x_2}$.
The intercepts are the points that cross the axes. The y-intercept is the point where the line crosses the y-axis and the x-intercept is where the line crosses the x-axis. Since the intercept is where the line crosses that coordinate's axis, the other coordinate is 0. That means the x-intercept is at "ground level" of the coordinate system, which means that it is not above ground (more positive than zero) nor below ground (more negative than zero). That means the y-coordinate at the x-axis is 0.
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| Since the point on the x-axis has not gone up or down when going to a, it's y-coordinate is 0. Since the point on the y-axis has not gone to the left or right when going to b, it's x-coordinate is 0. |
There are two similar ways to graph a line. The first is to choose two arbitrary numbers for x, plug them into the equation to find their corresponding y values, plot the points, and then draw a line that goes to infinity passing through those two points. The second is a specialized version of the first method. This second method is to find the two intercepts of the line, plot those points, and draw the line going to infinity through those two intercept points.
In these kinds of linear models, the slope is often referred to as the rate of change of y with respect to x. This is because the slope is a measure of how much y changes for every unit x changes. This means that this form is useful for modelling. For example, the rate of brain developement issues as a result of lead consumed can be described as 5 failed neuron connections per ppt of lead in the water. (The numbers are not true, but on average, the concept still holds true. Also, this is a simplified model.) So the model would be $y(n)=5 \frac{n}{ppt}*x(ppt)+0$.
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