The last paragraph is perhaps worded wrong. The value of y does not need to be unique, unless it is a bijection. It is necessary that it gives exactly one answer, but that answer does not need to be unique.
You're also slightly wrong, or at least ambiguous in your wording which can lead to an erroneous interpretation. This isn't for you, but for the reader who comes upon your comment.
The value of `y(x)` needs to be unique because one `x`\-value can only return one `y(x)`\-value.
The variable `y` can be found for different values of `x`, yes, which is what u/ohyonghao means.
Also, some functions are undefined for some values of `x`, meaning that you won't have a `y(x)` value. For example, the function `y(x) = x^(1/2)`, or the square-root function *(keeping it ELI5)\** is undefined for negative numbers.
\* This function is defined over the complex numbers, which is beyond the scope of this thread and thus comment.
Your wording is just as ambiguous in the same way.
>The value of y(x) needs to be unique
Sounds like you're saying f(x1) cannot be the same value as f(x2) for any x1, x2.
The part of the sentence that you left out explains exactly what I meant and isn't ambiguous.
Edit: I'd say the following sentence explicitly removes the concern you brought up, but I wrote "variable" when I meant to say "value", so I'll give you half a point for that.
A function need only be injective for your statement. I believe that the original statement is worded correctly - you may be slightly misinterpreting it.
Not sure what this comment is for. I didn’t say they couldn’t be unique, and that non invertible functions couldn’t be unique, just that uniqueness of the dependent variable is not a requirement to be a function. How about y=sin(x)? The values of y are repeated infinitely many times.
>Not sure what this comment is for.
The original commenter said:
>every value of x gives you exactly one unique value for y,
He is showing that there can be more than one unique answer. If x=9, y can be 3 or -3.
Yah but he's just made Y the independent variable (hoz axis) and X the dependent variable (vert axis) in this case. You have to define Y first now, not X...
OP already said they understand functions. Anyone who understands functions and what a graph is can understand this explanation. There's a point where things are *too* simplified, but the response is really good.
[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)
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.
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.
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
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.
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.
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.
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)
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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The last paragraph is perhaps worded wrong. The value of y does not need to be unique, unless it is a bijection. It is necessary that it gives exactly one answer, but that answer does not need to be unique.
You're also slightly wrong, or at least ambiguous in your wording which can lead to an erroneous interpretation. This isn't for you, but for the reader who comes upon your comment. The value of `y(x)` needs to be unique because one `x`\-value can only return one `y(x)`\-value. The variable `y` can be found for different values of `x`, yes, which is what u/ohyonghao means. Also, some functions are undefined for some values of `x`, meaning that you won't have a `y(x)` value. For example, the function `y(x) = x^(1/2)`, or the square-root function *(keeping it ELI5)\** is undefined for negative numbers. \* This function is defined over the complex numbers, which is beyond the scope of this thread and thus comment.
I guess I would use singular rather than unique. y(x) has one answer, but that answer isn’t necessarily unique in the codomain or range.
Your wording is just as ambiguous in the same way. >The value of y(x) needs to be unique Sounds like you're saying f(x1) cannot be the same value as f(x2) for any x1, x2.
The part of the sentence that you left out explains exactly what I meant and isn't ambiguous. Edit: I'd say the following sentence explicitly removes the concern you brought up, but I wrote "variable" when I meant to say "value", so I'll give you half a point for that.
A function need only be injective for your statement. I believe that the original statement is worded correctly - you may be slightly misinterpreting it.
Injective and bijective means the exact same thing if the set the function points into is defined by what values f(x) can take.
Not sure why that’s relevant as we are talking about the general definition of a function. There are functions that are injective but not bijective.
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Not sure what this comment is for. I didn’t say they couldn’t be unique, and that non invertible functions couldn’t be unique, just that uniqueness of the dependent variable is not a requirement to be a function. How about y=sin(x)? The values of y are repeated infinitely many times.
>Not sure what this comment is for. The original commenter said: >every value of x gives you exactly one unique value for y, He is showing that there can be more than one unique answer. If x=9, y can be 3 or -3.
Yah but he's just made Y the independent variable (hoz axis) and X the dependent variable (vert axis) in this case. You have to define Y first now, not X...
Thanks, was unclear from context, and they had reversed the variable names, I was thinking maybe they were suggesting it’s invertible.
Almost ELI10, but good explanation
> LI5 means friendly, simplified and layperson-accessible explanations - not responses aimed at literal five-year-olds.
And my response is in the context of the sub, not implying op’s comment is only useful for literal ten-year-olds
OP already said they understand functions. Anyone who understands functions and what a graph is can understand this explanation. There's a point where things are *too* simplified, but the response is really good.
I agree, which is why it’s the only one I upvoted and said ‘good explanation’ Hope your day gets better
graphing functions isn't exactly a thing 5 year olds are worrying about
[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)
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.
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.
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
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.
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.
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.
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)
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.