How do you graph and find the vertex for # y = 4 - abs(x+2) #?

1 Answer
Jun 13, 2015

Answer:

Take the graph of #abs(x)#, turn it upside down, shift it two to the left, and elevate it vertically by four, making the vertex at #(-2,4)#.

Explanation:

Sometimes it's best to write these equations in a slightly different way, to see it. Here, #y# can be rewritten:

#y=-abs(x+2)+4#

From there it also helps to know what the graph of #y=abs(x)# looks like:

graph{y=abs(x)}

Notice it's just the graph of #y=x# mashed together with #y=-x#. And like you would expect with #abs(x)#, there are no negative #y# values. And if we graph the negative, #y=-abs(x)#, we turn it upside down, like so:

graph{y=-abs(x)}

Adding 2 to the inside of the absolute value shifts the vertex (perhaps counter-intuitively) to the left by 2 units, like so:

graph{y=-abs(x+2)}

Finally, adding 4 to the end of our new graph, shifts the vertex 4 units vertically, creating the following:

graph{y=-abs(x+2)+4}

Finis!