Question

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

Answer 1

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!