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?
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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.
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