function Archives - Mathematics for Teaching

Prerequisite knowledge for calculus

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This post describes foundational reasoning abilities and mathematical knowledge students need to develop before beginning a course in calculus.

1. Covariational reasoning

Continue reading “Prerequisite knowledge for calculus” →

Algebra

How to Teach the Equation of Linear Function

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In my earlier post on Linear Function, I described how a linear function can be recognized based on equation, graphs, and tables. In this article, let us talk about how to derive the equation of linear function (as they are called in calculus) or the equation of a line (as they called in analytic geometry). I will be using linear function and line interchangeably in the discussion as most teachers would do. It is important to note and to make sure that students are aware that when the topic is on linear function and the teacher says “line” as in “What is the equation of this line?” it actually mean, “What is the equation of the linear function represented by this line?”

What is the equation of linear function?

Textbooks define a linear function as a function defined by the equation y=ax+b (or y=mx+b as m is commonly used for this form) or the equation ax+by+c=0. The latter equation is called the standard form of the equation of a line. In my opinion, this form should not be used when talking about equation of linear function because it does not show clearly the relationship between x and y. The equation y=ax+b shows the relationships between the independent variable x and the dependent variable y, where the value of the variable y is determined by the rule ax+b. But as I said earlier, in school mathematics, this is used interchangeably since we can transform one to the other by algebraic manipulation and we would be getting the same set of points from both. For example the equation 2x-y-3=0 could be transformed to y=2x-3.

Where did y=ax+b come from?

In my previous post about linear function I introduced linear function as a polynomial function of at most degree 1 so ax+b defines the linear function x?ax+b (also expressed as y=ax+b), where a and b are constant. In this case, linear function is defined based on the structure of the equation defining it. Do you think this would make sense to students? Telling is never a good way of teaching mathematics. I propose here a simple activity that would lead to the derivation of of the linear equation of a linear function from graphs.

Activity

The Cartesian plane is made up of points and each point is named by the ordered pair (x,y). In the figure on the right, there is something special about the points E, D, C, F, B, G, and A. They all belong to one line. They are collinear. What does it take for a point to be a member of this elite group? Can you just think of any point and say that it belongs to that group? Observe the x and y coordinates of the points on the line. What do you think would the coordinates of the middle point between D and C? How did you get that? What about the middle point between E and D? What would be their coordinates? If you think you have discovered the condition for membership on this line, try more points. Find the coordinates of the midpoints between C and F, F and B, B and G and G and A.

Challenge: Does the point P=(-12,-20) belong to the line where points A, B, C, D, E, F and G are?

Question to the teacher-reader: How would you proceed from here to derive the equation y=mx+b?

Next: How to derive the equation of linear function from its graph (This is the continuation of the Activity presented in this post.)

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.

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.

Algebra

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?