Functions and their graphs

A function assigns exactly one output to each input, so a graph that hits a given vertical line twice would be assigning two different $y$ values to the same $x$, breaking the "exactly one output" rule.

($\Rightarrow$) Suppose $C$ is the graph of a function $f$, so every point of $C$ has the form $(x_0,f(x_0))$ for some input $x_0$. Fix any vertical line $x=x_0$; the only points of $C$ that can lie on it are those with first coordinate $x_0$, and there is only one such point, $(x_0,f(x_0))$, because $f$ assigns a single value to $x_0$. So the line meets $C$ in at most one point.

($\Leftarrow$) Conversely, suppose every vertical line meets $C$ in at most one point. Define $f$ by: for each $x_0$ that has a point of $C$ above it, let $f(x_0)$ be the $y$-coordinate of that (unique, by hypothesis) point. This assigns at most one output to each input, so $f$ is a well-defined function, and by construction its graph is exactly $C$.

The two directions together show the equivalence: being a function's graph and passing the vertical line test are the same property viewed from two angles.

A constant slope between every pair of points is exactly the geometric definition of "straight" — the direction never bends.

Compute the rise and run between the two points on the graph: rise $=y_2-y_1=(ax_2+b)-(ax_1+b)=a(x_2-x_1)$, using the fact that both points satisfy the same rule $y=ax+b$.

Since $x_1\ne x_2$, we may divide both sides by the run $x_2-x_1\ne 0$: $$\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{a(x_2-x_1)}{x_2-x_1}=a.$$

This computation used only the values $x_1,x_2$ and the rule $y=ax+b$, and the $b$ term canceled out completely — so the ratio equals $a$ regardless of which two points $(x_1,y_1),(x_2,y_2)$ were chosen. A curve on which the slope between every pair of points is the same fixed number is, by the geometric definition of straightness, a line; hence the graph of $y=ax+b$ is a straight line of slope $a$.

Conversely, given any non-vertical line with slope $a$ passing through a point $(x_0,y_0)$, setting $b=y_0-ax_0$ produces a linear function $y=ax+b$ whose graph is exactly that line, which establishes the converse direction of the correspondence between linear functions and non-vertical lines.