How do you find the vertex of an absolute value function?

Answer 1

There are three main types of vertices for an absolute value function. There are some vertices which are carried over from the function, and taking its absolute value makes no difference. For example, the vertex of the parabola y = 3*x^2 + 15 is not affected by taking absolute values.

Then there are some vertices which are reflected in the x-axis because of the absolute value. For example, the vertex of the absolute value of y = 3*x^2 - 15, that is y = |3*x^2 - 15| will be the reflection of the vertex of the original.

Finally there are points where the function is "bounced" off the x-axis. These points can be identified by solving for the roots of the original equation.

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The above answer considers the absolute value of a parabola. There is a simpler, more common function, y = lxl. In this form, the vertex is (0,0).

A more general form is y = lx-hl +k, where y = lxl has been translated h units to the right and k units up. This function has its vertex at (h,k).

Finally, for y = albx-hl + k, where the graph has been stretched vertically by a factor of a and compressed horizontally by a factor of b, the vertex will be at (h/b,ak).

Of course, you can always find the vertex by graphing, especially since you might not remember the 2nd or 3rd parts above.