Question

What is the vertex of the graph of \[f(x)=\left| x-13 \right|+11\]?

Answer 1

Hint: We are given a question with a function we have to plot the given function and then we have to find the vertex. The given function has a modulus function, so we will be first opening up the modulus function and getting the corresponding positive and negative function based on the values of x. After opening up the mod function, we will get a pair of equations, which is, \[y=x-2\] and \[y=-x+24\]. We will then draw the graph of the given two equations. We can then figure out the coordinate of the vertex which is the intersection of the above obtained equations.

Complete step-by-step answer:
According to the given question, we are given a function whose graph we have to plot and then we have to find the vertex.
The given function we have is,
\[f(x)=\left| x-13 \right|+11\]-----(1)
If we see the given function, we can see that there is a modulus function. In order to plot the graph of the given function we have to open up this mod function. The modulus function has a special property, which is, rewriting the negative numbers to positive numbers.
So, we can the modulus part of the given function as,
\[\left| x-13 \right|=\left\{ \begin{matrix}
  & x-13 & x\ge 13 \\
 & -\left( x-13 \right) & x\le 13 \\
\end{matrix} \right.\]
For \[x\ge 13\], we get,
\[y=\left( x-13 \right)+11\]
Solving the above expression, we get,
\[\Rightarrow y=x-13+11=x-2\]
\[\Rightarrow y=x-2\]----(2)
Next, for \[x\le 13\], we get,
\[y=-\left( x-13 \right)+11\]
\[\Rightarrow y=-x+13+11\]
\[\Rightarrow y=-x+24\]----(3)
We will now plot the graphs of the equation (2) and (3). We will have the points of intercepts fro both the equations. And we get the graph as,

Here, we can see that the two lines intersect at the point \[\left( 13,11 \right)\].
Therefore, the vertex of the given equation is \[\left( 13,11 \right)\].

Note: Since the vertex of the given function is the intersection of the equation of lines obtained from the given function, we can find the point by substituting one equation into another. That is, we have,
\[x-2=-x+24\]
\[\Rightarrow x+x=24+2\]
\[\Rightarrow 2x=26\]
\[\Rightarrow x=13\]
Putting the value of ‘x’ in one of the equation, we get,
\[y=13-2\]
\[\Rightarrow y=11\]
Therefore, the coordinate of the vertex is \[\left( 13,11 \right)\].