Feb 142014
 
What is a linear function?

In mathematics, a linear function is used to name two different but related notions. In calculus and analytic geometry, a linear function is a polynomial with a highest degree of one. In linear algebra, the linear function is the linear map. This article is about the linear function in calculus and analytic geometry. This is the one we study in high school.

What does a polynomial with highest of degree 1 mean?

Will that include zero? Yes. That is why it would include what is also referred to as a constant function or the zero polynomial. Will that include algebraic expression with negative exponents or fractional exponents (they are also less than zero)? No. Because these expressions are not polynomials.

Will any polynomial of degree 1 qualify as a linear function?

Yes. For example if the polynomial of degree 1 has only one variable say 2x+3, then that defines a function x→2x+3. In symbol we can write this as f(x) = 2x+3 of if we let y=f(x) then we write y=2x+3. If the polynomial has several independent variable, say the polynomial 2x+3y+z, then it is the linear function defined by f(x,y,z)=2x+3y+z.

What does the graph of a linear function look like?

For the linear function in one variable, it is a line not parallel to the x-axis (inclined). For the linear function of degree zero, it is a line parallel to the x- axis. For the linear function with several independent variables, the graph is a hyperplane. In this post we will stick with the linear function in one variable. Examples of their graphs are shown below.

Graphs of linear function

Graphs of linear function

What is common about the two lines?

They are both lines, that’s for sure. However for both graphs, the change in y is the same for every unit of increase in x. If the coordinates are tabulated as shown below, we can see the increase/decrease in y stays the same or constant for every increase in x. The top table is for the red line and the bottom table is for the blue line. This is also how you can tell from the table of representation whether the relationship between x’s and y’s is linear or not. The change in y should be constant for per unit change in x.

calculating the gradient

The change in the value of y is constant per unit change in x

What do you call the ratio between the change in y vs the change in x?

If you look at the line as a representation of a function, we say that it is the rate of increase or decrease (also called rate of change). If you look at the line simply as a geometric figure, we say that it is the gradient or the measure of the slope of the line. Sometimes textbooks and teachers use this interchangeably. Since the slope refers to the change in y for every unit of increase in x, its formula is

formula for slope

How to calculate the gradient

Can we consider all lines as representations of linear function?
vertical line

This is not a linear function

Take a look at the line on the right? Does it have the same slope? If you calculate it using any two points, you will get k/0. The number is undefined. You could argue that the value of the slope is still the same anywhere only that it is undefined. Alright.

Is it a function? No. Remember that a function is a relationship between the x and the y values such that for every x, there is one unique y value.

Coming up next: How to teach the equation of a linear function.

 

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