The graph of the linear equation is, as it's name says, a straight line. But why the points of quadratic, cubic, etc, functions are connected by a smooth curve instead of a line? I've heard that the reason is because if we use straight lines to connect the points of a quadratic function, that wouldn't show the true behavior of the function. What does this mean, exactly?
On the other hand, I have read that there exist functions which doesn't have graphs. What are examples of these functions?
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For your OTHER question:
You want to remember that a graph is a visual representation of a function.
For example, let's say my function turned "Bananas" into $3$, and "Apples" into $6$.
Most likely here, I'd represent my function as a table.
Here, you speak of functions that take a subset of $\mathbb R$ (domain) and map it onto another subset of $\mathbb R $ (range).
In this case, say we have a function $f$, then we would represent the function on a coordinate plane as all points that are of the form $(x,f(x))$.
If you had a function that took in one number, let's say: time, and gave two outputs, say temperature and humidity: you would need three axes to plot a point on (time, temperature, humidity).
You can still represent this, but not on a coordinate plane, for certain.
Once you have more than three dimensions, it's hard to make a non-table visual model.
You sometimes have to introduce representations like vector fields (arrows) and colors, to make the data stick on a $2d$ plane or a $3d$ model.
When you say 'the points of the function', I imagine you are considering values at integers, or maybe a few other nice numbers that you plug in. So for $y=x^2$ you would have points $(1,1),$ $(2,4)$, $(3,9),$ $(-1,1)$, $(-2,4)$ , etc... maybe a few others. However, to use same terminology as what you heard, the "true function" has infinitely many points. You could put $(1/2,1/4),$ $(1/4,1/16),$ $(3/4,9/16),$ $(-\sqrt{2},2)$ and all kinds of points down for any $x$ value you can think of and they all have to lie on the graph of the function.
In fact, there are so many points that need to lie on the graph that they are infinitely close together. Thus there is really no freedom in how you draw the lines 'connecting the points'. The way the graph needs to curve coming out of a point is totally determined by the points infinitely close to it. If you plot points on a dense grid of $x$ values you will see them blend together into a curved shape. Then you can just draw a smooth line through them and it will be curved in the shape of a parabola (or whatever function you're graphing). This is just what a graphing calculator does when it draws the graph of a function. (It might even draw straight lines in between the points, but the points are so close together that your eye can't tell the difference.