Why is there $(2,3)$ corner coordinates on the graph, using this function: $f(x)=2x-1+|x-2|?$

Question

Currently I am studying about absolute values and I had to consider this function and draw it's graph:

$$f(x)=2x-1+|x-2|$$

I helped myself with symbolab, here is the link: https://www.symbolab.com/solver/step-by-step/f%5Cleft(x%5Cright)%3D2x-1%2B%5Cleft%7Cx-2%5Cright%7C/?origin=button

I understood everything about getting the $x$ coordinate $(-1, 0)$ and $y$ coordinate $(0,1)$ but I don't see where did $(2,3)$ corner coordinates come from or what does it mean - could anyone please tell that to me$?$

I am also including the picture: graph and it's coordinates

Answer 1

Since $$|x - 2| = \begin{cases} x - 2 & \text{if $x \geq 2$}\\ -(x - 2) & \text{if $x < 2$} \end{cases} $$ the function $f(x) = 2x - 1 + |x - 2|$ can be defined piecewise as \begin{align*} f(x) & = \begin{cases} 2x - 1 + x - 2 & \text{if $x \geq 2$}\\ 2x - 1 - (x - 2) & \text{if $x < 2$} \end{cases}\\ & = \begin{cases} 3x - 3 & \text{if $x \geq 2$}\\ x + 1 & \text{if $x < 2$} \end{cases} \end{align*} Thus, if $x > 2$, the graph has slope $3$, while the graph has slope $1$ if $x < 2$. The graph has a corner at $x = 3$ since the slope to the right of $x = 2$ is different from the slope to the left of $x = 2$.

Answer 2

$(2,3)$ is clearly a point on the graph because it satisfies the equation:

$$3=2\cdot2-1+|2-2|$$

The reason why the graph has an angle at that point is because it is the minimum of the second part of the expression, $|x-2|.$ Compare this to the minimum of the function $|x|$ itself at $x=0.$