How do you find the vertex???

[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 didn't really know how to make it space right

 

[Mrspi]

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 didn't really know how to make it space right

I'm not sure about what you're being asked to do...but here's how I would think about it.

You should know what the graph of y = |x| looks like. It's a v-shaped graph, opening up, with the vertex at (0, 0)

Let's take your first problem....

if you have y = |x - 2| the vertex of the graph will be moved 2 units to the RIGHT...when x = 2, y = |2 - 2|.... or y = 0. So you know that the x-coordinate of the vertex is 2.

You know this now....if y = | x -2 |, the vertex of the graph is at (2, 0).

And what happens if you multiply the right side of this equation by -1, to get

y = -|x-2|

Well, that will CHANGE the sign of each y-coordinate you got when you dealt with y = |x - 2|. This will have the effect of reflecting the graph over the x-axis (turning it "upside down")

Finally, there's that "-12"...adding -12 to each y-coordinate moves the vertex of the graph DOWN 12 units, from (2, 0) to (2, -12)

If y8u actually graph this function, you may be better able to understand this explanation.

 

[Mrspi]

I guess what I was saying in my previous LENGTHY answer is that you should think about what the basic graph looks like, in this problem,

y = | x |

and consider what happens to that graph if you add or subtract something from x, as in

y = | x +/- a |

and then what happens to the vertex if you add or subtract something FROM the absolute value expression...like,

y = | x +/- a | +/- b

 

[Mackenzie25]

I understand the concept of moving the vertex based on the location of the numbers you are adding and/or subtracting (y = | x +/- a | or y = | x +/- a | +/- b)
However, my teacher would like us to solve for the vertex without physically (or mentally) graphing it. I'm kind of having difficulties with this, because the work has to be shown, and there has to be a system of equations similar to the ones that I attempted in my first post. I don't really know how to set up the equation, so this is where I'm really having trouble.
Thanks for your response, it really helped me figure out how to define and explain that better on future problems

 

[mmm4444bot]

I'm working on graphing absolute value equations …

… my teacher would like us to solve for the vertex without physically (or mentally) graphing it …

:roll:

Why would a teacher want to restrict students from thinking graphically?

?

… there has to be a system of equations …

I don't understand the "system" at work in your first post.

The following is somewhat contrived, but it does result in a system of equations, the solutions to which are the coordinates of the vertex.

|x - 2| by definition is either x - 2 or 2 - x, depending upon the value of x.

y1 = -(x - 2) - 12

y2 = -(2 - x) - 12

These simplify to:

y1 = -x - 10

y2 = x - 14

Since the vertex occurs at the intersection of these sets of ordered pairs, we know that y1 = y2, so just call them both y.

y = -x - 10

y = x - 14

There's your system of equations.

Journey onward!

Personally, if I were instructed to find the vertex coordinates without graphing anything (physically or mentally), then I would skip the system of equations.

Go back to |x - 2| = x - 2 or 2 - x, depending upon the value of x.

Obviously, at the intersection (vertex), |x - 2| is both.

x - 2 = 2 - x

Solve for x.

x = 2

Substitute x = 2 into the expression for y, to get the other vertex coordinate.