Using Slope-intercept Form
By Kathleen Cantor, 03 Apr 2021
The slope-intercept form is the most common linear equation format. Although this form, written as
y=mx+b, can look odd and confusing due to its use of variables, it is easy to use with proper explanation and practice. Even more, you can extract vast amounts of information from a single equation.
As mentioned previously, slope-intercept form, much like other forms used to model functions, uses variables to describe the meanings behind values in an equation. In order to use the slope-intercept form effectively, it is crucial that you understand and can determine what these variables mean for the equation as a whole.
Arguably the most important variable in the slope-intercept form is x. This is the input of the equation. You could say that x is the value that goes into a "machine" (which is, in this case, the function) to produce another value. For example, say you have the equation
2x = ? which is the "machine" in this case. When you plug x into the equation as a value, say, for the purpose of this example, 2, you would replace x with 2 and proceed to do 2 times 2, getting 4. The x value that you plugin can be any number, positive or negative.
Now, just as the function "machine" has an input, it also has output, modeled by the variable y. Unlike x, you typically do not decide what the y value is. Instead, you allow the x value to determine what the output will be. Say that someone presents you with another function,
3x + 1 = y. Notice that there are now two variables with y replacing the question mark of the previous example. When you chose an x value, for example, 3, you would plug this in and do
3(3) + 1 to get 10, which is the output, y.
To show this input-output relationship between two values, you would write x and y values as something called an ordered pair. Ordered pairs simply put the x value (input) and y value (output) of a particular function together, formatted as (x, y). In the equation and example above, you would write the ordered pair as (3, 10) per the format.
The ordered pair changes with the x value, as changing the input, in turn, changes the output. This means that for every function, there are numerous unique ordered pairs. Ordered pairs are important when it comes to plotting a function on a plane, which is a surface of two intersecting perpendicular lines, one horizontal and one vertical. On this plane, ordered pairs act as directions for the person graphing the equation. The x telling them to go that many units right (or, if x is negative, left), and y telling them to go that many units up (or, if y is negative, down) when starting at the axis (where the two lines of a plane intersect).
The m variable, known as the "slope" in the slope-intercept form, is more complex compared to the other variables. Simply put, the slope is the rate at which the y values of a function grow or decrease in relation to the x values. In other words, m describes how a function's ordered pairs change. You can find a line's slope or function by examining the rate of rise (how much the y value changes between two points) to run (how much the x value changes between those same two points). You do this by using the equation
(y2 - y1)/(x2-x1)=m where (x1, y1) and (x2, y2) are ordered pairs of a function.
For example, if you were given two ordered pairs, (3, 5) and (5, 8), of a single function, the equation would look like
(8-5)/(5-3). When this is simplified, 3/2 is produced as the answer and the slope, m, for the function. Later, we will discuss the slope's importance and how you can use it in the slope-intercept form.
Of course, there is still the question of the last variable, b. This variable is the constant in the slope-intercept form, meaning that it is an unchanging number in the equation compared to the variables x and y. The main job of the variable b is to tell you, at a glance, what y equals when x is 0, known as the y-intercept.
The y-intercept is found by simply plugging in a 0 for x in whatever function someone gives to you. For example, if the function is
y = 4x + 3, you plug in 0 to get
y = 3 as the y-intercept (the ordered pair would be (0, 3)). Think of the b value as a shortcut to finding the y-intercept; instead of replacing x with 0, you just have to remember that (0, b) is the y-intercept.
With a basic understanding of slope-intercept form variables, you can now use and apply these concepts to find information and create graphs.
Finding the x Value Using Slope-Intercept Form
Although we mentioned that you wouldn't normally be the one to decide the value of y, there are cases where you will need to. Normally, you use this method when you want to find the x value given a particular y value. Using the slope-intercept form, this is a fairly easy thing to do if given the function and y-value. For instance, if you are presented with the function
y = 6x - 1 and are told to find x when y is 11, you would plug in y, giving you 11 = 6x - 1. Then after adding 1 to each side of the equation and dividing by 6, you would get
x = 2.
Graphing Functions Using Slope-Intercept Form
As mentioned previously, ordered pairs (x, y) produced by inputting values for x and getting an output, y, are used as directions for points on a plane. When you have multiple ordered pairs (two or more), you can plot these points and connect them to present what is known as a linear function, which is a straight line going through all of the points in a function. Linear functions are capped on either side by arrows, which indicate that the line goes on forever in both directions.
Slope and Using Slope-Intercept Form
The slope is a key component of using slope-intercept, specifically when trying to figure out how a graph will look plotted on a plane. If you understand slope and its meaning, with a brief interpretation of m, it is possible to determine the line of a function.
A general rule when examining the slope is that the larger m is, the steeper the line will be. This makes perfect sense. The output, y, increases faster, even when x grows at the same rate. For example, if you have two equations,
y = 2x and
y = 6x, and x is 1 for both, y in the first function would be 2 (ordered pair would be (1, 2)) but, for the second, the y value would be the greater value, 6 (ordered pair would be (1, 6)). If you graphed both of these functions, you would see that y = 6x is far steeper because the y values increase faster.
When m is positive, as seen in the examples used thus far, it is a general rule that the line produced from the function will appear to increase as the x value increases (gets larger). The opposite is generally true when m is negative: the line will appear to decrease as x increases. This information is extremely important when you are trying to quickly determine how your graph will look without finding ordered pairs and/or graphing the function.
The Importance of the Slope-Intercept Form
Slope-intercept form not only gives you the ability to find ordered pairs but also allows you to form a general idea of what your graph will look like with minimal effort. As long as you understand the variables and what they mean in the context of a graph, you can do almost anything using the slope-intercept form.
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