How do you graph # f(x)= abs(x-6)#?

1 Answer
Aug 31, 2016

I like graphing using piece-wise functions.

Let's write two piecewise functions, one for the area of the function where #y# is decreasing, another for the part where #y# is increasing.

The function will change from a decrease to an increase at the vertex. When given an absolute value function of the form #y = a|x - p| + q#, the vertex is at #(p, q)#.

Since #p = 6# and #q= 0#, #f(x) = |x - 6|# has its vertex at #(6, 0)#.

The next step is to determine the range. This will be determined by two elements:

a) The y-coordinate of the vertex
b) The direction of opening

In the function #y = |x - 6|#, the vertex has a y-coordinate of # 0# and the function opens upward (since parameter #a# is positive).

Hence, the range of this function is #y ≥ 0#.

Now that we know the vertex, we can find another point on the graph and then find the equation of both piece-wise functions. I think it may be simplest to find the y-intercept of the function.

#y = |x - 6|#

#y = |0 - 6|#

#y = |-6|#

#y = 6#

Hence, the y-intercept is at #(0, 6)#. Start by finding the slope between the two points we found.

#m = (y_2 - y_1)/(x_2 - x_1)#

#m = (6 - 0)/(0 - 6)#

#m = 6/(-6)#

#m = -1#

We need to find the equation using point slope form now.

#y - y_1 = m(x - x_1)#

#y - 6 = -1(x - 0)#

#y - 6 = -x#

#y = -x + 6#

This is the left-hand side equation, since #y# is decreasing due to the negative slope. You will graph this line paying attention to the range, #y ≥ 0# (you must stop the line once you hit the line y = 0#).

As for the right-hand side piecewise equation, this can be obtained by multiplying the #mx +b# side of the left-hand piece-wise function by #-1#.

#y = -(-x + 6)#

#y = x - 6#

In summary, our piecewise equations to graph are #y = x - 6, x ≥ 6# and #y = -x + 6, x < 6#.

You should have gotten a graph similar to the following.

enter image source here

Hopefully this helps!