# why do we need to study linear function

A linear function is any function that graphs to a straight line. What this means mathematically is that the function has either one or two variables with no exponents or powers. If the function has more variables, the variables must be constants or known variables for the function to remain a linear function. To identify linear functions, you can create a checklist of several items the function must meet. The first item the function must satisfy is that it must have either one or two real variables. If another variable is present, it must be a known variable or constant. For example, the function

C = 2 * pi * r is a linear function because only the C and r are real variables, with the pi being a constant. The second item is that none of the variables can have an exponent or power to them. They cannot be squared, cubed, or anything else. All variables must be in the numerator. The third item is that the function must graph to a straight line. Any kind of a curve disqualifies the function. So, linear functions all have some kind of straight line when graphed. The line could be going up and down, left and right, or slanted but the line is always straight. It doesn't matter where on the graph the function is plotted as long as the line comes out straight. If you know that a function is linear, you can plot the graph using just two points. If you are unsure, you can use three or four points to double check. To figure out your points to plot them, set up a T-chart and start plugging in values for one of the variables. Plug the values into the function to calculate the other variable and note it on the T-chart. When you have filled the T-chart, go ahead and plot the points on the graph.

Then, take a ruler and make a straight line through them. All linear functions will have points that are lined up nicely. Let's try graphing y = 2 x. First, we fill up a T-chart. I'm going to do four points so you can see how the points line up and how, if you know that the function is linear, just two points are sufficient. Now, that I've filled up my T-chart, I can go ahead and plot them. Let's see what we get. The linear function is arguably the most important in mathematics. It's one of the easiest functions to understand, and it often shows up when you least expect it. Because it is so nice, we often simplify more complicated functions into linear functions in order to understand aspects of the complicated functions. Unfortunately, the term linear function means slightly different things to different. Fortunately, the distinction is pretty simple. We first outline the, which is the favorite version in higher mathematics. Then, we discuss the, which is the definition one typically learning in elementary mathematics but is a rebellious definition since such a function isn't linear. In one variable, the linear function is exceedingly simple. A linear function is one of the form $$f(x)=ax,$$ where the $a$ is any real number. The graph of $f$ is a line through the origin and the parameter $a$ is the slope of this line. A linear function of one variable. The linear function $f(x)=ax$ is illustrated by its graph, which is the green line. Since $f(0)=a times 0 =0$, the graph always goes through the origin $(0,0)$. You can change $f$ by typing in a new value for $a$, or by dragging the blue point with your mouse.

The parameter $a$ is the slope of the line, as illustrated by the shaded triangle. One important consequence of this definition of a linear function is that $f(0)=0$, no matter what value you choose for the parameter $a$. This fact is the reason the graph of $f$ always goes through the origin. By this strict definition of a linear function, the function $$g(x) = 3x +2$$ is not a linear function, as $g(0) ne 0$. Why this insistence that $f(0)=0$ for any linear function $f$? The reason is that in mathematics (other than in elementary mathematics), we don't define linear by the requirement that the graph is a line. Instead, we require certain properties of the function $f(x)$ for it to be linear. One important requirement for a linear function is: doubling the input $x$ must double the function output $f(x)$. It's easy to see that the function $g(x)$ fails this test. For example, $g(1)=5$ and $g(2)=8$, which means that $g(2) ne 2g(1)$. We can write this requirement for a linear function $f$ as $$f(2x)=2f(x)$$ for any input $x$. If $f(x)=ax$, then $f(2x)=2ax$ and $2f(x)=2ax$, so this requirement is satified. To satisfy this doubling requirement, we must have $f(0)=0$. This follows from the fact that if you double zero, you get zero back. Therefore, the doubling requirement means $f(0)=2f(0)$, so $f(0)$ is a number that is the same if you double it; i. e. , $f(0)=0$. By the way, for a linear function, this property must be satisfied for any number, not just the number 2. A linear function must satisfy $f(cx)=cf(x)$ for any number $c$.

The other requirement for a linear function is that applying $f$ to the sum of two inputs $x$ and $y$ is the same thing as adding the results from being applied to the inputs individually, i. e. , $f(x+y)=f(x)+f(y)$. The rebellious view of the linear function is to call any function of the form $$f(x)=ax+b$$ a linear function, since its graph is a line. An affine function of one variable. The affine function $f(x)=ax+b$ is illustrated by its graph, which is the green line. Since $f(0)=a times 0 +b=b$, the graph always goes through the $y$-axis at the point $(0,b)$, which is illustrated by the gray point. You can change $f$ by typing in a new values for $a$ or $b$, or by dragging the blue points with your mouse. The parameter $a$ is the slope of the line, as illustrated by the shaded triangle. However, as mentioned above, this type of function with $b ne 0$ does not satisfy the properties for linearity. So, to call $f$ a linear fuction, we have to rebellious ignore such facts to the contrary. Strictly, if $b ne 0$, then $f$ should be called an affine function rather than a linear function. Given that this rebellious view is firmly entrenched in elementary mathematics, we might sometimes join in and use this terminology. If it doesn't seem worthwhile to insist on the distinction, we might use the term linear function when we should really use the term affine function. In other contexts, the properties of linearity are critical for the mathematical analysis. In such cases, we'll be careful to insist that a linear function $f(x)$ does satisfy that $f(0)=0$, and make the distinction between linear and affine functions.

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