Graphs of the parabola, exp and log

To understand why a graph is a straight line or it's a curve you don't need to understand calculus, but you need to understand the tangent. This page here is meant to give a very rough idea of the graphs. I'm not going to talk about inflection points, critical points and derivatives here.

I'm also making the assumption that you understand the unit circle and you already know how to plot one point of a function. All you need to understand to plot one point of a function is to understand that each point is an ordered pair in the form [math]\displaystyle{ (x, \ f(x)) }[/math], with [math]\displaystyle{ f(x) }[/math] being the vertical axis.

In case you ask about the word "slope". I'm using slope, rate of change and tangent interchangeable here.

The parabola

Have you ever asked yourself why is [math]\displaystyle{ f(x) = x^2 }[/math] a parabola? If [math]\displaystyle{ f(x) = x }[/math] is a straight line, how can the square be so different? I'm going to explain this with a pure geometrical reasoning.

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Look at the graph of [math]\displaystyle{ f(x) = x }[/math]. Every point of this function has this form [math]\displaystyle{ (x, y) }[/math] with [math]\displaystyle{ x = y }[/math]. Now the meaning of this equality in the graph is that for every step that you take in [math]\displaystyle{ x }[/math], the same step is taken in [math]\displaystyle{ f(x) }[/math]. In other words, one unit forwards is equal to one unit up and vice-versa. Take any two points where [math]\displaystyle{ b \gt a }[/math]. We have that [math]\displaystyle{ \large\frac{f(b) - f(a)}{b - a} = 1 }[/math]. See how the rise [math]\displaystyle{ f(b) - f(a) }[/math] and the run [math]\displaystyle{ b - a }[/math] are equal to each other? We have a name for the ratio rise / run and it's //tangent//. Do you remember from school that the tangent of 45° is 1? Right now you may be unaware of the definition of a derivative but we've just discussed the concept of the rate of change, which is the geometrical idea of a derivative.

Why is [math]\displaystyle{ f(x) = x }[/math] a straight line? Because its rate of change never changes! It's always constant and always equal to 1.

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Now look at [math]\displaystyle{ f(x) = x^2 }[/math]. Take two points where [math]\displaystyle{ b \gt a }[/math]. For example: [math]\displaystyle{ f(1) = 1 }[/math] and [math]\displaystyle{ f(2) = 4 }[/math]. Now take another two points elsewhere, [math]\displaystyle{ f(3) = 9 }[/math] and [math]\displaystyle{ f(4) = 16 }[/math]. In the first case the rate of change is 1/3. In the second case the rate is 1/7. Every different pair of points that you choose is going to result in a different rate of change. This "proves" that the graph cannot be a straight line, because the rate of change between every pair of points is always different from the rate of another pair. For every step we take in [math]\displaystyle{ x }[/math], [math]\displaystyle{ f(x) }[/math] is going to be the square of the same distance. In other words, as we walk at a constant speed along the x axis, from 0 to infinity, each point of the graph is going higher and higher at even faster speeds. From 0 to negative infinity the same function's behaviour.

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At school you may have learnt that the more points you plot, the more precise your graph is. How does a computer plot a graph? Back at school you may have had already noticed that when the points are very close to each other, connecting two points next to each other with a straight line did resemble a curve. That's how a computer plots a graph, it traces straight lines but because the points are so close to each other, we can't distinguish on screen a dot from a line. Right know you may be unaware of the definition of a limit, but to plot infinitely many points close to each other is the concept of a limit. Computers can't plot infinitely many points though because there isn't infinite time and because we don't need to. Also, screens with infinite resolution doesn't exist.

= image graph_high_power.png width="25%" /=

Now what happens with large, positive and even powers? It's a parabola, but we "flatten" it. The reason for this behaviour is quite simple. For [math]\displaystyle{ 0 \lt x \lt 1 }[/math] we have fractions and the denominator grows very large with very large powers. When [math]\displaystyle{ x \gt 1 }[/math] the power is so large that the function grows fast enough to become closer to a vertical line.

What about cubics and odd powers? The concept is the same of a parabola with just one difference. For [math]\displaystyle{ x \lt 0 }[/math] the graph goes down rather than up. Remember that the minus sign is kept with odd powers.

Note: In case you noticed that the above graph is slightly distorted compared to the real graph of a biquadratic function for example. I used inkscape to draw a bezier curve and because the software is unable to handle polynomials with a degree higher than cubics, the distortion that you are seeing is exactly the error associated in attempting to trace a biquadratic function using a cubic to approximate it. I've just explained in practical terms one of the fundamental problems that numerical methods have to solve, which is to approximate functions and minimize errors. _

The exponential and logarithm

= image graph_exp_log.png width="50%" /=

Take [math]\displaystyle{ a^b }[/math] with [math]\displaystyle{ a, \ b \gt 1 }[/math], for example [math]\displaystyle{ 2^2 }[/math]. You can easily notice that for every unit that you add to the power, the result is even larger. In other words, the rate of change is always changing and thus, the graph cannot be a straight line.

What happens if [math]\displaystyle{ a \lt 0 \lt 1 }[/math]? Then, if we keep increasing the power, it results in smaller and smaller numbers. It's an exponential that decreases for larger powers. The graph is not a straight line because the rate of change keeps changing as we move along it.

What happens if [math]\displaystyle{ a\gt 1 }[/math] and [math]\displaystyle{ 0 \lt b \lt 1 }[/math]? Then, from school, we have a root and the graphs of roots are also not a straight line.

The definition of a logarithm relies on the definition of the exponential. When we add the concept of functions we learn that the log and the exp are one the inverse function of the other.

Note: the same comment as above. I used bezier curves to trace the graphs of exponentials and logarithms. It's impossible for a polynomial to perfectly match the exp or log.

The sine and cosine waves

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(they aren't perfect sines or cosines because they were hand drawn) /=

Why are they waves? You need to understand the unit circle and how to read it. First, angles as we learn in euclidean geometry, are always in between 0° and 360°. That is, from zero to doing a full circle turn. Any angle beyond that is just multiples of full turns or partial turns. With negative angles not meaning "an angle that is less than nothing" but just reflecting the fact that we can measure angles clockwise or anti-clockwise. This also explains that we are not required to plot the graph beyond one full turn, because anything beyond that is just repeating the same pattern.

At school is common to teach how to measure angles first, then the relationship between angles and ratios between sides of a right triangle. Lastly, trigonometry is combined with the definition of functions. By definition, sines and cosines map a measure, the angle, to a ratio between sides of the right triangle. Both are nothing more than numbers, irrational numbers most of the time. That's what a trig function is. Any number such as 1000 or 0.3 is treated as multiple turns, half turns, partial turns, etc along a circle for the purposes of trigonometric functions. That's why sines and cosines are waves that keep oscillating from 1 to -1 back and forth.

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Choose a point anywhere on the perimeter of the unit circle. Use it to draw a right triangle with one of the vertexes being the center of the circle. Notice that whenever you move the point on the perimeter clockwise or anti-clockwise, the ratio between any two sides of the triangle keeps changing. This "proves" that the sine and cosine are not "zig-zag" graphs, but smooth waves.

Note: In case you know just a bit of numerical methods and wondered "Can I trace waves with parabolas?". Yes, that is possible. But here is the problem, one parabola is never a wave. What do you do? Break the wave into multiple segments, each one being the segment of a parabola. So you have a series of parabolas, half with upwards concavity, half with downwards concavity. Parabolas can never perfectly fit the curvature of a sine or cosine wave though.

An extra comment to add to the above. Sometimes people make the connection between the sine and cosine waves with the shape of the circle itself. Careful! The curvature of the sine and cosine waves are not that of half-circles! A parabola can be distorted such that it fits a half-circle but to understand that, one has to understand some concepts from analytical geometry. I'd theorize that this confusion comes from the fact that the unit circle can, erroneously, be seen as the graph of a function itself. The graph of a function can, at best, trace half of a circle. Most textbooks in calculus mention the "vertical line test" to differentiate graphs of functions from graphs of other equations which aren't functions.

The tangent

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= image graph_tangent.png width="90%" /=

Let's move [math]\displaystyle{ p }[/math] to a position that is associated to the angle 45°. Just by doing this we get two values "for free", without having to calculate anything at all. [math]\displaystyle{ \theta° = 45° }[/math] because, in Euclidean geometry, the sum of all three angles of a triangle must be always 180°. The other value that we have is [math]\displaystyle{ a = r }[/math]. We know that we have a triangle with two angles equal to 45°. The only way for this to happen is to have two sides, excluding the hypotenuse, equal to each other. The line that is tangent to the circle at [math]\displaystyle{ p }[/math] is unique, there is only one and it happens to coincide with the side [math]\displaystyle{ a }[/math]. There another fact in the triangle that we have. The ratio [math]\displaystyle{ r/a = a/r = 1 }[/math] happens to be the length of the side [math]\displaystyle{ a }[/math]. Now, [math]\displaystyle{ r }[/math] is a constant and equal to one, it's much easier to calculate [math]\displaystyle{ a/1 = a }[/math] for every angle than the inverse of that.

Now move [math]\displaystyle{ p }[/math] anti-clockwise and close to the null angle (but keep the tangent perpendicular!). What happens with [math]\displaystyle{ a }[/math]? Its length becomes closer to zero. Move [math]\displaystyle{ p }[/math] in the other direction (but keep the tangent perpendicular!), clockwise and closer to the right angle. What happens with [math]\displaystyle{ a }[/math]? Its length stretches so much that if we reach the right angle, [math]\displaystyle{ a }[/math] goes to infinity. Hence, tangent for 90° doesn't exist. Another way to see this fact: what is the tangent to a perfectly plane surface? It would be a line that is parallel to the plane (we are talking about trajectories which are straight lines here, no parabolas), but in this case either the line is contained in the plane or doesn't touch it at all. Hence, this contradiction shows that tangent of 90° doesn't exist.

With sines and cosines we saw that the sides of the triangle never go beyond 1 unit, the triangle is always inside the unit circle. With tangent, however, the triangle has sides crossing the unit circle. Therefore, the graph won't be a wave limited by 1 and -1. For angles very close to 90° and 270° the graph is going to extend to almost vertical lines. For angles close to 0° or 180° it's going to be a curve, not a straight line, until it reaches zero. It's a function that associates the angle with the length of [math]\displaystyle{ a }[/math]. That's nothing new because sine and cosine are defined as relationships between angles and lengths of triangle's sides.

= image graph_tangent3.png width="30%" /=

In case you wondered, there is another way to see the relationship between angles, sides and the tangent. Let's look at the same triangle that we had for sines and cosines and think on the ratio rise / run. For 45° it's pretty easy to see that rise = run. For 0° we have that [math]\displaystyle{ \sin(0) / \cos(0) = 0 }[/math]. For 90° we have that [math]\displaystyle{ \sin(\pi/2) = 1 }[/math] and [math]\displaystyle{ cos(\pi/2) = 0 }[/math] and we can't divide by zero. In other words, tangent is also a sin / cos ratio.

After you understand the three basic trigonometric function's graphs, extend the same reasoning for the inverse trigonometric functions. The concept shouldn't be hard to grasp. It's the inverse of what we did just now. The inverse trigonometric functions associate the lengths or ratios, with the angle. That is, the input is the ratio or length, the output is the angle. _

Hand traced graphs

Depending on how hard it is to trace graphs by hand for you, heavily distorted graphs can very well lead to incorrect interpretation. I don't know if there are teachers who are pedantic with the "correctness" of graphs. One of such cases is sine and cosine. They both have identical shapes. The sole difference is that sine and cosine differ in that [math]\displaystyle{ \sin(0) = 0 }[/math] and [math]\displaystyle{ \cos(0) = 1 }[/math], whereas [math]\displaystyle{ \sin(\pi/2) = 1 }[/math] and [math]\displaystyle{ \cos(\pi/2) = 0 }[/math]. Plotting the graph without attention to this fact can lead to mixing up one with the other and causing calculations to go wrong.