Graph y = |x – 4| + 2.

What can we tell about this function at a glance? We're no Sherlock Holmes, but we can see that the vertex is at (4, 2), and that the graph will open up because a = 1.

Now we need to do a little legwork. Just one more point will do, and then we can use symmetry to finish this graph off. Usually we're a big fan of picking x = 0 as our go-to point. That isn't the best idea here, though.

The farther we get from the vertex on the x-axis, the larger our values of y will be. Let's stay close with x = 5.

y = |5 – 4| + 2

y = 1 + 2 = 3

When x = 5, y = 3. We can use this to get a third point as well, using the symmetry of absolute value functions to our advantage. We know (5, 3) is one point to the right of the vertex; if we go one point to the left of the vertex, we'll also have y = 3. So (3, 3) is our third point on the graph.

Graphing absolute value functions is elementary, dear Shmooper.

Write an equation of the graph shown.

The vertex is (0, 3). This means the beginning of the equation is:

y = a|x – 0| + 3

y = a|x| + 3

What is a? "The first letter of the alphabet" is true, but unhelpful in this case. To find the value of a, we can plug in a point from the function into our equation so far and solve. The point (1, 5) will do nicely.

5 = a|1| + 3

a = 2

Tada! The equation for the graph is y = 2|x| + 3.

Graph .

The function's vertex is (-1, -3). And a = is positive, so the graph points up. We also know that the graph will be wider than y = |x|, because a is less than 1.

Are we a smarty-pants know-it-all, or what? Well, not yet at least. We still need a few points for plotting. Let's pick an x-value close to the vertex; how about our old buddy x = 0?

So the point is on our graph. We can use the absolute value's symmetry to find another point: x = 0 is one step to the right of the vertex, so one step to the left of the vertex (x = -2) will also equal .

Looking at our graph, we totally could have picked x = 4 to multiply the fraction away, making everything a zillion times easier. Confound our hindsight, letting us see our missteps with such clarity.