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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So a graph will have an x-axis and a y-axis right?
And with a function you have a x, and a y or f(x) (those are the same thing just different names)
A graph shows you all the (x, y) pairs.
So like for y=x if you plug in 1 for x you get 1 for y and so on.
For the graph you will get a straight like that’s at a 45 degree angle. Because you are plotting all the places where x is the same as y.
Going from function to graph is generally a lot easier than graph to function.
For function to graph plug in a bunch of numbers and just mark them down on a graph.
But for graph to function you tend to need to think a bit and having some knowledge of basic function shapes is a must have. As well as how parts of a function will shift the graph. I could go into more detail here if you want, but its verging on a whole topic overview.
A function will usually be expressed as an equation that sets a variable, _y_, to some expression including another variable _x_.
On a graph, _x_ is the _independent variable_. It’s the one whose value you get to change just because you want to.
The other variable, _y_, is the _dependent variable_. You don’t set that one directly, you can only find out what it is by setting _x_ to a value and solving the expression.
On a graph, _x_ goes on the horizontal axis. When you pick a value of _x_ and solve it, put a dot for _y_ at the right distance above or below the horizontal axis for that value of _x_. If the function is _y = x_, and you pick 2 for _x_, you put a dot 2 units above the x axis at 2 units to the right of where the horizontal and vertical axes cross.
If you do that for all possible values of _x_, you will get a shape on your graph. For _y = x_, it’s a straight line ascending at 45° right through the origin (where the axes cross).
Part of the definition of a function is that every value of _x_ gives you exactly one unique value for _y_, so you will never see a line double back above the horizontal axis, it only moves up & down across all values of _x_.
Imagine you have a function like f(x) = x^2.
You can think of the graph of the function as all the pairs of numbers: (x, f(x)). In the specific case of this function, that’s the pairs of numbers (x, x^2).
For example, (2, 4) and (3, 9), and (67.23, 4519.8729) are all points in the graph of the above function.
If you consider all pairs of numbers (x, f(x)), you get the graph of the function. Take a plane with an x-axis, and a y-axis, and plot the pairs of numbers: x on the x-axis, and f(x) on the y-axis.
Now obviously you can’t plot every pair of numbers since there are an infinite amount of them, but what you can do is take some of the integers values and plot them. In the case of (x, x^2), we have:
(-4, 16), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16) and so on.
Then draw a curve that goes between these points, and you have a visualisation of the graph of a function. The actual graph is all values (x, x^2), however.
In fact, the graph of a function is sufficient to describe the function completely!
If the line on the graph is a straight line, then we read it from left to right, is the line is going up, left to right, it is a + equation, if it is going down, it is a – equation.
We look at where the line crosses the vertical axis, is is the y axis. What ever value that is, we write it down.
From that y-intercept, we count to the right how many spaces until the line crosses a whole number, then we count how many spaces vertically this covers as well. This gives us the slop of the line, rise over run. Often thos slop will be expressed as a fraction like 2/3. Meaning we have 2 spaces of rise for every 3 spaces to the right of run.
So, if we see that the line is going down from left to right, the y intercept is 7, and the rise over run is counted to be 1/2, we can then plug those values into an equation. Y=mX+b
Y is the answer we are often trying to find, m is the slope we have figured out, X is usually the given variable that may change, and b is where the line intercepts the y axis.
In my example the equation thay represents the line observed is y= -1/2(x) + 7
With this you could easily determine what value of why falls on the line for a given value of x
[When I was learning I just plotted points until I saw how these graphs were being generated. Once you know these few (especially the first 6) the rest (ie higher degrees and polynomials) are just variations.](https://nohemiportfiolio2012-2013.weebly.com/uploads/1/9/2/5/19257411/748647836_orig.jpg?245)
When a function is described as “y = [something] x” or “f(x) = [something] x”, that’s just a way of saying “if you fill in x and do [something] with it, you get the matching y value. You can do this by hand with a few x-y pairs, draw them in a coordinate system, and connect the literal dots — that’s a graph right there.
It can help you visualize the variable y for each input of x.
A function is a y value generating machine, you plop in a value for x and the machine returns a value for y. If you are plotting a simple line, the y values are easy to predict. When you get to functions that have a square or a cube or is an absolute value (whatever), things get a lot more interesting. When we, in match class, say something like “this function is symmetrical about the y axis), what we are saying is that for each negative x value the corresponding positive x value will produce the same y value. So your x value of -2 will produce a y value of 2, x value of 2 will also produce a y value of 2.
The graph of a function helps tell us “what is this function *doing?”* as the x values go from negative infinity to infinity.
You pick a point on the x-axis (the horizontal one), this point will be your x value, you go upwards (or downwards) until you bump into the graph, then from the meeting point you go to the left or to the right until you bump into y-axis, this next point will be our y value.
When you have some function y=f(x) (like y=3x+2, an easy one because it is a line), you have to assign values to x and figure out what y value matches it (when x=2, y=8), and then you go onto your handy-dandy graph paper and put a dot at the place that has that specific x and y value. Do this for a number of different x values and see what sort of pattern develops.
The reverse idea, getting the applicable formula from looking at a graph, is partly recognition of the pattern (certain functions make certain shapes) and learning how constants in a function affect where the graph sits in x-y space. That is a bit more complicated but most of the common functions like lines, parabolas, circles, ellipses, can be taken from a graph and written out as a formula.
In our case, if we had plotted that example line, we would have a line rising from left to right (a positive slope, the y value gets bigger as x gets bigger). We could get the slope pretty easy (change in x and change in y between two points) and see that it is 3. We get the y intercept=2 by looking where the line crosses the y axis at x=0 (when X=0, y=2).
It turns out that a lot of functions, even the higher order ones, have inflection points (places where the graph changes direction or hits a minimum or maximum) and these locations are defined by the constants in the function, so you have a pretty good idea where you want to graph the function in x-y space to show its overall shape (not much point in plotting the function y=x^2 +3 way out where x=500, for example, unless that is the region that really interests you for other reasons; it won’t tell you that the minimum y value happens when x=0).
EDIT: it gets a lot more difficult when you are dealing with more than two variables (higher dimensions than x versus y) but the same general rules apply. Drawing it can become difficult.
The function is just an instruction for putting dots on a rectangle of “graph paper”. For any position in the left/right direction, it tells you where the dot goes in the up/down direction. (Try it–plug in whatever value of x you want and it will tell you the value of y.)
If you have infinite dots, you get a line (straight or curved)–and that’s the graph of the function.
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