I can interpret functions in the formula form but I simply can’t understand how it can become a graph. How would you transform one into the other?
In: 46
It’s something you need to practice with, having someone explain it will probably never really get the idea into your head. My best advice would be to find some interesting functions and then create a [function table](https://youtu.be/oK47UCT2dl0) (AKA an input-output table) then fill in the table using some numbers and plot those numbers on the graph. Eventually, after doing many of them, you will see how certain operations shape the graphs. When you work through it, you should do your best to find interesting points, most notably where the function equals 0, but also where the function has its maximum and minimum points, where the direction of the function changes, where the function has undefined values, etc. You should also always try to use different “types” of inputs, such as negative numbers, irrational, fractions, even complex if you really want to get into it.
Here are a few functions you can start with (also, always remember that you can think of “f(x)” as the “y” variable. f(x) goes on the vertical axis and x goes on the hori
f(x)=5x-2
f(x)=x^2 + 7
f(x)=e^(x)
f(x)=3sin(5x – 2) + 4
f(x)=ln(x)
Making a graph is really handy to “see the big picture” of a formula. The example we’ll use is y=x*2.
| If x is | Y would be |
|—|—|
| 0 | 0 |
| 1 | 2 |
| 1.5 | 3 |
| 2 | 4 |
| 3 | 6 |
A graph basically does the same thing as that table, but for more numbers. Start with a dot at “0 steps the the right” and “0 steps up” (0,0). Then the next dot goes at “1 step to the right, 2 steps up” (1,2). Then a dot at “1.5 (total) steps to the right, and 3 (total) steps up” (1.5,3). Another dot at (2,4) and a fifth dot at (3,6).
That’s how you plot just those exact answers, but what if we want more dots than that? Well, you can do the math for what y would be when x is 2.5, and 3.5, and 2.75, and 2.6373847… and if you did the math for every little infinite number, eventually all the dots would be touching and form a line. **The magic of graphs is that if you just make a few dots, you can usually eyeball drawing the line, without having to do infinite math.** If you plot just that table, you should notice that the dots are all in a line. And if you try to draw a nice straight line connecting all of those dots, you’ll find that if you do the math for x=2.5 (2.5,5), the dot ends up being on the line you drew!
Going from the graph to the function is trickier, it mostly breaks down to recognizing the overall shape of the graph, straight lines, vs lines with 1 curve vs lines with two curves, etc. And then using some exact numbers to work backwards.
One value is the input (usually x) and another value is the output (usually y). This can also be applied to more than 2 dimensions if you use multiple inputs.
Imagine you have a machine called f. you throw number in f and f gives you some number out(when possible).
When we write f(x), we mean that what we’re throwing inside is x.
For any point in the graph, the x value of the point shows what is the number we put inside, while the y value of the point shows what is the number we get out.
On a basic level a function (lets call it f) takes an input (lets call it i) and gives you an output (lets call that o).
A function only takes certain inputs, so lets put all of these allowed inputs into a set (lets call that set X).
Now we can put each individual i out of our set X into the function f and we reciev a certain output in relation to our input, so lets call this output o_i (i in reference to the input i)((so if we put in a differen i from our set x (lets call it j) we would get the output o_j)).
Now we can write this: f(i)=o_i
Now we can describe a graph in simple terms:
A graph is a set which contains tupels of the form (i,o_i).
This set (=graph) of our function f contains all possible touples regarding each element in X.
In a more math-like notation:
f: X -> f(X) and
Graph of f: {(i,f(i)}
For illustration let’s assume we have function in the form
y=f(x)
We select a suitable vector of values for x, perhaps
X=1:100
Calculate the corresponding y values.
Baby LP will
Plot y against x (with x by convention on the horizontal axis)
Now we can read of values of y for anq
A function tells you a relationship between two values. For example, the function y=2x tells you that y is twice the size of x.
A graph shows you how that relationship looks for a range of values. The process of making a graph is simple, you put a range of values into the function and plot the resulting values against the values you put in.
Most of the time you will look at a graph to understand the “rate of change” which is how the relationship (not the values themselve) changes as you change the values. For example, if I double X does Y double too? And does this happen no matter what value of X I start with? If the graph is a straight line then you know it will, but if the graph curves you know that the behaviour of the function changes based on the value of X you have.
Another use of graphs is to quickly see what the biggest and smallest values your function can make are.
Basic case: listplots.
Take the function f(n)=2n which doubles numbers. You only allow n to be whole numbers. So 1->2, 2->4, 3->6 and so on. You would like to visualize this somehow, so you get a piece of paper and:
– draw a horizontal line and mark 1,2,3,4,…
– At the marking for 1 go up 2 and mark that point, because 1->2.
– At the marking for 2 go up 4 and mark that point, because 2->4.
– At the marking for 3 go up 6 and mark that point. Go on for a few more points.
– We say you have marked the points (1,2), (2,4), (3,6),…
You should see points along a straight line. The mathematical term “graph” refers to the collection of points { (1,2), (2,4), (3,6),…}, but slightly imprecisely also the picture you just drew (also called a “plot”). Given a function, you can draw the graph just by marking points. Given a drawing of the graph you can read of the values of the function. That is all.
Continuous case:
You now have a function f(x)=x^2, which computes squares and allow x to not just be whole numbers, but also f(0.5)=0.25 for example.
You again get a piece of paper and want to do the same thing:
– Draw a horizontal line and mark 0,1,2,3,4… . You call it the “x-axis”.
– For each value on the x-axis, you go up the amount dictated by the function and mark that point, e.g. (2,4), (3,9), (0.5, 0.25), (0.1, 0.01). Note that here you also need to plug in “x” which are not just whole numbers.
– Problem: You need to do this for many, many, many points.
So, in principle, nothing changed here. You still have a function and can draw the graph just by marking points. You can also read off the function, e.g. f(0.5)=0.25, by seeing that (0.5,0.25) is on the graph.
However, you cannot easily draw thousands of points (though computers can). So what you learn in school is how to “cheat” to get out of all that work. Namely, you notice that for a quadratic function like this all points lie on a curve, which you call parabola. So you mark five or ten points and then just draw a curved line that looks about right. You made a sketch, good enough.
Next step: You are now given f(x)= 3 x^2 – 6x +5 instead and are asked to draw a graph.
– Hm, that is quadratic, so it should “look like” a parabola.
– Option 1: Mark about 20 points by plugging in different values for “x” and make your sketch this way.
– Option 2: You notice that f(x)=3(x-1)^2+4. So this is just a shifted parabola, rescaled by a factor 3. So you mark the point (1,4) and draw a parabola centered at this point, which is 3 times steeper than in the previous example. Done. This is usually what schools want you to do.
At its basic level, the graph is plotting the relationship between the input (x) and the outputs (f(x)) for the function. To start with, the best thing is to make a table of values for x and then calculate the corresponding f(x) values and plot them.
Unfortunately, that’s tedious, so we don’t de it every time. Luckily, there are function families with specific shapes that are easy to start to recognize. You can see a cat and identify it as a “cat” without all cats needing to look identical, because you understand some basic properties of cats and how they’re different from, say, raccoons or dogs. Same thing with functions. A function in the form of f(x) = x^(2) will look different than f(x) = 1/x, which looks different than f(x) = sin (x), but functions in the form f(x) = ax^(2) + c will have some very similar characteristics.
Once you learn about the various function families, you can learn about some common transformations. Adding a constant term “outside” the function will shift the graph up/down (f(x) = sin(x) + 2). Adding a constant term “inside” the function (f(x) = sin(x-3) ) will shift the function left/right. Multiplying “outside” the function (g(x) = -3sin(x)) stretches and/or reflects the function vertically. Multiplying “inside” the function ( g(x) = sin(1/2 x) ) stretches it horizontally.
Once you become familiar with the common transformations and the common function families, you can predict what a function looks like pretty easily without graphing technology. This is sort of like someone telling you about a cat that is an overweight, long haired, orange tabby with a white belly. That’s a pretty specific cat, but you can most likely picture it in your mind if you’re familiar with cats. With practice, algebraic functions become similar.
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