polynomial functions Archives - Mathematics for Teaching

Making Sense of Power Function

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The power function, $ax^n$ , n = positive integral exponents is actually the ‘basic’ polynomial function.They are the first terms in the polynomial function.

With graphing utility, it is no longer as much fun to graph function. What has become more challenging is interpreting them. Here’s are a set of tasks you can ask your learners as review for function. You can give it as homework as well.

Consider the sets of power function in the diagrams below. Answer the following based on the diagram

What are the coordinates of the points of intersection?
Why do all the graphs intersect at those points?
When is $x^4 < x^2$ ?
When is $x^7 > x^3$ ?
Why is it that as the degree or exponent of x that defines the function increases, the graph becomes flatter for the interval -1<x<1 and steeper for x > 1 or x >-1 ?
Sketch the following in the graphs below: $t(x) = x^{10}$ , $l(x)=x^9$
Why is it that power function with even exponents are in Quadrants I and II while power function with odd exponents are in Quadrants I and III? Why are they not in Quadrant IV?

power function with even exponents — Power function with even exponents

power function with odd exponents — Power function with odd exponents

What other questions can you pose based on the graphs above? Kindly use the comment section to suggest more questions. Thanks.

My other posts about function

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.

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.

Algebra, Assessment

What are the big ideas in function ?

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Function is defined in many textbooks as a correspondence relationship from set X to Y such that for every x (element of X), there is one and only one y value in Y. Definitions are important to know but in the case of function, the only time students will ever use the definition of function as correspondence is when the question is “Which of the following represents a function?”. I think it would be more useful for students to understand function as a dependence/covariational relationship first than for them to understand function as a correspondence relationship. The latter can come much later. In dependence/co-variational relationship “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (Sfard, 1991, p. 15)

The concept of change and describing change is a fundamental idea students should learn about functions. Change, properties, and representations. These are the big ‘ideas’ or components we should emphasize when we teach functions of any kind – polynomial, exponential, logarithmic, etc. Answer the following questions to get a sense of what I mean.

1. Which equation shows the fastest change in y when x takes values from 1 to 5?

A. y = 4x² B. y = -2x² C. y = x² + 10 D. y = 6x² – 5

2. Point P moves along the graph of y = 5x², at which point will it cross the line y = 5?

A. (5, 0) and (0,5) B. (-5, 0) and (0,-5) C. (1, 5) and (-1, 5) D. (5, 1) and (5,-1)

3. Which of the following can be the equation corresponding to the graph of h(x)?

A. h(x) = x³ + 1 B. h(x) = x³ – 1

C. h(x) = 2x³ + 1 D. h(x) = 2x³ + 4

4. The zeros of the cubic function P are 0, 1, 2. Which of the following may be the equation of the function P(x)?

A. P(x) = x(x+1)(x+2) B. P(x) = x(x-1)(x-2) C. P(x) = x³ – x²D. P(x) = 2x³ – x² – 1

5. Cubes are made from unit cubes. The outer faces of the bigger cube are then painted. The cube grows to up to side 10 units.

The length of the side of the cube vs the number of unit cubes painted on one face only can be described by which polynomial function?

A. Constant B. Linear C. Quadratic D. Cubic function

Item #1 requires understanding of change and item #5 requires understanding of the varying quantities and of course the family of polynomial functions.

Of course we cannot learn a math idea unless we can represent them. Functions can be represented by a graph, an equation, a table of values or ordered pair, mapping diagram, etc. An understanding of function requires an understanding of this concept in these different representations and how a change in one representation is reflected in other representations. Items #2 and #3 are examples of questions assessing understanding of the link between graphs and equations.

Another fundamental idea about function or any mathematical concept for that matter are the properties of the concept. In teaching the zeroes of a function for example, students are taught to find the zeroes given the equation or graph. One way to assess that they really understand it is to do it the other way around. Given the zeroes, find the equation. An example of an assessment item is item #4.

You may also want to read How to assess understanding of function in equation form and Teaching the concept of function.

Algebra, Assessment, High school mathematics

Algebra test items – Graphs of rational functions

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TIMSS (Trends in international Math and Science Study) classifies test items in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

In my earlier post about this topic on using the TIMSS Assessment Framework for constructing test items I presented a set of questions about zeros of cubic polynomial function. Here are three more test items about graphs of rational function based on the framework. Note that questions should be independent of each other, that is, an answer in one item should not serve as clue to the other items. I only used the same rational function here to highlight the differences among the cognitive domains – knowing, applying, reasoning.

Knowing

What may be the equation of the graph below?

Applying

The graph above the x-axis is function f and the graph below the x-axis is function g. Which of the following equations describes the relationships between f and g?

a. g(x) = f(-x) b. g(x) = f^-1(x) c. g(x) = f^-1(-x) d. g(x) = -f(x) e. g(x) = /f(x)/

Reasoning

Carlo drew the figure below by graphing two functions on the same coordinate axes. The graph on the left is f(x) = 4/x². Which of the following function is represented by the other graph on the right (the blue one)?

a. $g(x)=\frac {4}{x^2}$ b. $g(x)=4+\frac {4}{x^2}$ c. $g(x)=\frac {4}{(x-2)^2}$ d. $g(x)=\frac {4}{(x-4)^2}$ e. $g(x)=\frac {4}{(x+4)^2}$