Tag: linear functions

Posted in Algebra

What is a Linear Function?

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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.

 

Posted in Algebra

What is an algebraic function?

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An algebraic function is a function created by applying the operation of addition, subtraction, multiplication, division, and extracting the nth root. Let me give an example. Suppose you have the function f and g where f is a linear function and g is a constant function.  Let f(x)=x and g(x) = -3. We can create another linear function h by multiplying f and g that is h(x) = -3x. We can also create another linear function l where l = f – g, that is l(x) = x-3.

What about quadratic functions? A quadratic function (with real roots) is a product of two linear functions. So we can make a quadratic function n by multiplying f and l for example. That is, n(x) = f(x) x l(x) = x(x-3). And cubic function? A cubic function is a product of three linear functions or of a quadratic function and a linear function. And quartic function? Well, you must have figured it by now. This process of creating function by multiplying linear functions produces a family of functions called polynomial functions so called because its algebraic representation is a polynomial.

functions
Polynomial Function Family

What kind of function is produced when you divide a function by a function in x? Using the function defined earlier, what is g÷f?  g÷l? f÷l? Getting the quotient of two polynomial functions give us a new family of functions: p(x) = -3/x; q(x) = -3/(x-3); and, r(x) = x/(x-3). These expressions defining the functions will not simplify to polynomial expressions so they do not belong to the family of polynomial functions. They belong to what is called the family of rational functions so called because they are defined by rational expressions.

You can also raise a function to a fractional power, that is get the nth root of the function. For example we can have t(x)= x^0.5. That is t(x)=sqrt of x. I don’t know what this family of function is called. Maybe we can call then nth root functions.

These three families — polynomial functions, rational functions, and nth root functions, all belong to the family of algebraic functions. Functions that are not algebraic functions are called transcendental functions.

You may also want to read ideas for teaching functions.