Mistakes regarding graphs

• Crescent, decrescent and constant functions. First, for languages that are written right to left this may be a source of confusion in comparison to languages that are left to right. The other is about reading the graph itself. It my happen that some people associate in their minds "positiveness" with a function being crescent and "negativeness" with being decrescent. While "null" would mean constant. Yet another source of confusion is to think that concavity up or concavity down are, respectively, synonyms for crescent and decrescent. I'm only speculating, but some people may even make the association between a ramp and acceleration and confuse crescent with decrescent and vice-versa because of that.
  
  I think this misconception comes from the way teachers at school teach functions. A function is crescent in a certain interval not because all values of [math]\displaystyle{ f(x) }[/math] in that interval are positive, but because the rate of change is positive. A function is decrescent in a certain interval because its rate of change is negative, not because [math]\displaystyle{ f(x) }[/math] lies within the negative part of the vertical axis. Lastly, being constant means a zero rate of change, not literally [math]\displaystyle{ f(x) = 0 }[/math].
  
  Can we say that a function is crescent, decrescent or constant on a point? This question is conceptually imprecise. A function is crescent or decrescent over an interval, a sequence of points. Without an interval there can't be a rate of change to begin with.

• Suppose that the graph of a function displays [math]\displaystyle{ f(b) \gt f(a) }[/math] with [math]\displaystyle{ b \gt a }[/math]. That does not mean that the function is always crescent from [math]\displaystyle{ a }[/math] to [math]\displaystyle{ b }[/math]. The graph may have points where it changes from crescent to decrescent and vice-versa. For the same reason [math]\displaystyle{ f(b) \lt f(a) }[/math] doesn't guarantee that the function is always decrescent in that interval.

• There is a common mistake that really demonstrates that people don't understand the concept of a function at all. Suppose that a problem describes something such as the population's growth, the variation in temperature in respect to time or the brightness of a light in respect to electrical power. The problem gives the graph, which excuses us from having to gather data and then finding the function. Now the question is something such as: beginning at 2 hours, after 1 hour, what is the temperature going to be? For a brightness of x units, what is the electrical power? What's the expected population at year x? Not knowing to answer these questions while the graph is given means that people really don't understand functions at all. Now the curious fact is that maybe the same people are able to do the calculations and solve the equations.

• Reading the graph of a single variable function is pretty easy. Nonetheless, from time to time, some people confuse [math]\displaystyle{ x }[/math] with [math]\displaystyle{ f(x) }[/math]. This is specially problematic with physics. Suppose we have a function that depicts Speed x Time. It's not uncommon for one to read the graph mixing up time with speed, pointing out the wrong values when asked about time or speed. When we learn about derivatives and integrals it becomes even worse! It's not uncommon for people to confuse the graphs of each one because then we have not just one function, but three and three graphs.

• The other common mistake is to confuse position or trajectory with velocity or acceleration. Suppose that the function is a straight line that begins at some time and v(t) = -10 m/s. At some later time it's v(t) = 10 m/s. First, who said that the trajectory is a straight line? The graph is Velocity x Time. Second, velocity is a vector, the sign indicates the direction, not the magnitude. Third, by integrating v(t) we obtain x(t) and the graph should be parabola. However, who said that x(t) is a graph of a trajectory? The motion itself is not the parabola! It's the measure, the distance from a certain origin, that is behaving as a parabola.

• Trajectories x plane curves x graph of a function. It's not uncommon for people to confuse equations of lines or planes with functions of two or three variables.
  
  Suppose a person is running in circles on the Cartesian plane. That circle is described by an equation of a circle, like this: [math]\displaystyle{ 4 = x^2 + y^2 }[/math]. It's a circle with radius = 2 with the center at the origin (0,0). Later, when we learn about functions of two variables, that same equation is called a level curve. This [math]\displaystyle{ f(x,y) = x^2 + y^2 - 4 }[/math] is a function of two variables, not the equation of a circle!
  
  Careful with [math]\displaystyle{ y = x }[/math] and [math]\displaystyle{ y = f(x) }[/math]. The former can mean two things: that two numbers are equal to each other or the equation of a line, in which the coordinates of all its points obey to that equality. Now the latter is a way to say this "the vertical coordinate of each and every point of the function's graph is given by calculating the function at that [math]\displaystyle{ x }[/math]".
  
  To clarify the confusion between the XY Cartesian plane and the "X x f(X)" plane I'd say this: the XY plane has two independent axis where we can freely choose any number we want from one axis and from the other to find positions, points. That's why we can draw circles on it. Y does not depend on X and vice-versa. On the other hand, to plot functions, the vertical axis is always dependant on the values of the horizontal axis. That's why functions can only increase, decrease or remain constant. We either go forwards or backwards. If we "move" X to the right, f(x) goes along with it. The same for "moving" to the left. What ties [math]\displaystyle{ x }[/math] to [math]\displaystyle{ f(x) }[/math] so tightly? The function itself!

• Graph vs function. Is the graph of a function the function itself? Not quite. The function is a concept, it's not a drawing! The graph is a way to visualise the function to allow us to identity patterns. What can happen in our world is that some objects or shapes may resemble the graph of a function. Trajectories for example. A straight line is just that, a straight line. Now a linear function, the graph of it is a straight line. The line itself is not a function. We often use one or the other interchangeably but under some cases it's good to know the difference.

• This mistake arises from the fact that when we first learn about modulus, we don't know what a function is yet. Many teachers and books say that "modulus erases the minus sign" of a number. It's better to say that the modulus is a function, because modulus is really a piecewise function. So we have that [math]\displaystyle{ |2| = |-2| }[/math]. That definition "modulus is an operation that removes the minus sign" can lead us to think that "minus two is equal to two". When the correct way to read it is "The modulus of minus two is equal to the modulus of two". A very subtle difference! Nobody sees the expression [math]\displaystyle{ -2 = 2 }[/math] as true because it isn't. However, the consequence of misreading the modulus is wrongly plotting graphs of functions. Suppose that we have this function [math]\displaystyle{ f(x) = |x| }[/math]. When we think that "minus two is equal to two" we can, inadvertently, plot 2 and -2 at the same position. What is equal there is [math]\displaystyle{ f(-2) = f(2) \ !!! }[/math]. It's a quirk that happened to myself more than once.
  
  I've never liked the definition that modulus erases the sign of a number. We all learn that numbers are either positive, negative or zero. What is the "absolute number"?? Something that is neither positive nor negative? This is the confusion that comes along absolute value when we are unaware of functions.

• When drawing the unit circle, do not do this: [math]\displaystyle{ \cos(x) \times \sin(y) }[/math]. The unit circle is not a function of one variable and even less a function of two variables. It's a circle on the XY Cartesian plane. The axes themselves aren't functions nor angles. If you name the horizontal axis [math]\displaystyle{ \cos(x) }[/math] what you are saying is this "When [math]\displaystyle{ x = 1 }[/math], length is one and [math]\displaystyle{ \cos(1) = 1 }[/math]". Analogously, you are saying this for the vertical axis "When [math]\displaystyle{ y = 1 }[/math], length is one and [math]\displaystyle{ \sin(1) = 1 }[/math]". This mistake means you are, inadvertently, treating sine and cosine as linear functions.

• There is a simple confusion that must have happened to almost everyone when plotting trigonometric functions for the first time. When we first learn sine, cosine and tangent we have this strong geometric idea of angles and triangles. The plane has two axes, one for the variable, the angle in this case, and the other for the function's image. I'm pretty sure that almost everyone have, at least once, thought that, for sin(45°) for example, we go to the XY plane and literally measure an angle of 45° to plot a point. This happens way before we even learn about the polar coordinates.

• When we do [math]\displaystyle{ f(x,y) = c }[/math], a confusion that can happen is to think that [math]\displaystyle{ c = x = y }[/math]. We aren't making the function's variables constants, what we are regarding as constant is the image. What we are plotting is every pair [math]\displaystyle{ (x,y) }[/math] for which the function has a certain constant value. In the same way more than one [math]\displaystyle{ x }[/math] can map to one, and only one, [math]\displaystyle{ f(x) }[/math]. Multiple values for the dependent variables can result in the same constant with multivariable functions.