Why does this function ${f(x)} = {x^2}$ produce curved shape line on the graph? 
Guys, I'm learning algebra. And I'm very curious about this function's graph. Could you please tell me why the SQUARING FUNCTION ${f(x)} = {x^2}$ or ${g(x)} = {x^2}$ produce curved shape (Right side graph: Graph - Curved Line) line instead of straight line like left side graph (Graph - Straight Line) in the picture? On the left side I have connected all the points with small small straight lines. But, it is not correct according to book. There is no point discussed about the curved shape graph. Please enlighten me with your simple and expert opinion. Thanks in advance!
 A: When you draw the graph of the function you can find some points, like you have done, choosing some numbers $x$, calculating their images $f(x)$ and drawing this points $(x,f(x))$.
You have done this with $(0,0),(1,1),(2,4),(-1,1),(-2,4)$.
To have the correct graph of the function you should take EVERY different $x \in \mathbb{R}$ and draw every single point $(x,f(x))$.
This is obviusly impossible but more points you find and draw, more the graph that you draw will look like the original.
In this case, connecting the points you have drawn with a straight line is wrong because if you take other points, like $(\frac{1}{2},\frac{1}{4})$ you'll notice that this point is not on the straight line connecting $(0,0)$ and $(1,1)$ but it's below.
You generally have to draw as much point as you can and then connect them as smoothly as you can.
A: The way you produce the graph to the left is by taking a finite list of points, figuring out what $y$ values are generated by your $y=x^2$ curve to the right. Then you're just connecting all of the points with line segments. 
The curve on the right though takes every point in x and uses $y=x^2$ to produce a $y$ value. 
So the flaw is that you are thinking of the graph on the left as the graph of $g(x)=x^2$; it isn't. It's actually a piecewise function, and for the segment from $x \in [0,1]$, the function is defined as the following:


*

*$g(x)=x^2$ for $x \in \{0,1\}$, and 

*$g(x)= x$ for $ x \in (0,1)$. 


Notice that, for the first definition, $\{0,1\}$, is a set while for the second, $(0,1)$ is an interval. 
You might want to define the rest of the function for practice! 
A: You have only calculated the output values for some inputs. That is when $x$ is an $integer$. But the function is defined for the $ entire $ number line. Including  non-integral rational and irrational numbers . So you have to calculate the value of $x^2$ for those values as well .
For example $ (0.5 )^2$ is $ 0.25$ but according to your graph $ 0.5^2 $ is $0.5$ ! Which is why your graph is only correct for integral points.
A: In the first graph only a fiew points are connected, while in the second graph many points are connected. In a perfect ideal graph all the infinite points would be connected.
When you draw a straight line you only need to connect two points since all the others are automatically connected being on the line you drew. While in a parabola(for example the graph of  ${f(x)} = {x^2}$) the direction is ALWAYS changing, and you can never have three points or more on the same line. So if you draw graphs by connecting points trough straight lines, unless your graph is a straight line, you always can only make approximations of the ideal graph, for which you would need to connect all the infinte points. Computers can connect many points, infinite to our eyes, so the second graph you posted is very similar to the ideal graph.
A: Interestingly enough, the process you used to draw the graph is the basically the process used by a graphing calculator in drawing a graph: plot a bunch of points and connect them with straight line segments. The only difference is that the calculator can plot a lot more points, and more quickly, than you can. If you look carefully at the graph of $y=x^2$ on a graphing calculator, you will see this is the case. Now $y=x^2,$ or $f(x)=x^2$ is an example of a smooth function, in the sense that it has a derivative at each point on its graph (equivalently, a unique tangent line at each point on the graph). So, the graph obtained by plotting points and joining them by line segments (what we might call piecewise linear) fails to have a tangent at the points where the line segments meet, since the line segments have different slopes. That's the gist of it anyway. I could be more precise, but it would take longer. Also, I'm not sure of your mathematical background.
