# Thread: How do you find the vertex???

1. ## How do you find the vertex???

I'm working on graphing absolute value equations, and for one of my problems, I need to identify the vertex using two seperate system of equations; one for the x-coordinate, and one for the y-coordinate. I don't really understand how to do it, and my first attempt gave me the wrong answer. Does anyone understand how to do this? Any help would be appreciated.

Equation #1: y=-|x-2|-12

My work:
x-coordinate ...............y-coordinate
x=-|(0)-2| ..................y=-|x-2|-12
x=-|-2| .......................y=-|(-2)-2|-12
x=-(2) ........................y=-|-4|-12
x=-2 ...........................y=-4-12
...................................y=-16

I know that the y-coordinate is wrong, but I dont understand how to do the problem.

Equation #2: y=|2x+4|+3

My work:
x-coordinate .......................y-coordinate
x=|2(0)+4| .........................y=|2x+4|+3
x=|0+4| ............................y=|2(4)+4|+3
x=|4| ................................y=|8+4|+3
x=4 ...................................y=|12|+3
..........................................y=15

My apologies for all of the periods, I couldn't get it to space right.

2. Hello mackenzie25
Originally Posted by mackenzie25
I'm working on graphing absolute value equations, and for one of my problems, I need to identify the vertex using two seperate system of equations; one for the x-coordinate, and one for the y-coordinate. I don't really understand how to do it, and my first attempt gave me the wrong answer. Does anyone understand how to do this? Any help would be appreciated.

Equation #1: y=-|x-2|-12

My work:
x-coordinate ...............y-coordinate
x=-|(0)-2| ..................y=-|x-2|-12
x=-|-2| .......................y=-|(-2)-2|-12
x=-(2) ........................y=-|-4|-12
x=-2 ...........................y=-4-12
...................................y=-16

I know that the y-coordinate is wrong, but I dont understand how to do the problem.

Equation #2: y=|2x+4|+3

My work:
x-coordinate .......................y-coordinate
x=|2(0)+4| .........................y=|2x+4|+3
x=|0+4| ............................y=|2(4)+4|+3
x=|4| ................................y=|8+4|+3
x=4 ...................................y=|12|+3
..........................................y=15

My apologies for all of the periods, I couldn't get it to space right.
The vertex will occur where the expression inside the absolute value symbols, $\displaystyle |...|$, has the value zero. From this, you can work out the value of $\displaystyle x$, and hence the value of $\displaystyle y$.

So for equation #1:

$\displaystyle |x-2|=0$

$\displaystyle \Rightarrow x = 2$

Plug this value of $\displaystyle x$ into the equation for $\displaystyle y$:

$\displaystyle y = -|2-2| -12$

$\displaystyle \Rightarrow y =0 -12$

$\displaystyle \Rightarrow y = -12$

So the vertex is at $\displaystyle (2,-12)$

And for #2:

$\displaystyle |2x+4|=0$

$\displaystyle \Rightarrow 2x=-4$

$\displaystyle \Rightarrow x = -2$

And so $\displaystyle y = |2(-2)+4|+3$

$\displaystyle \Rightarrow y = |0| +3$

$\displaystyle \Rightarrow y = 3$

So the vertex is at $\displaystyle (-2,3)$