SOLUTION: Here's my question: Sketch the graph of each function or relation and state the domain and range. Y = │x│ - 4 I can figure it out on the graphing software once I

Algebra ->  Graphs -> SOLUTION: Here's my question: Sketch the graph of each function or relation and state the domain and range. Y = │x│ - 4 I can figure it out on the graphing software once I      Log On


   

Question 241542: Here's my question:
Sketch the graph of each function or relation and state the domain and range.
Y = │x│ - 4
I can figure it out on the graphing software once I figure out the problem. I am not sure how to figure out a problem like this. Can someone help me please?
Thanks.
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
y+=+abs%28x%29+-+4
There are several way to graph this:
  • If you know what the graph of y+=+abs%28x%29 looks like then the graph of y+=+abs%28x%29+-+4 will be the graph of y+=+abs%28x%29 shifted 4 units down (because of the -4).
  • Use the fact that when x+%3E=+0 abs%28x%29+=+x and when x+%3C+0 abs%28x%29+=+-x. So for y+=+abs%28x%29+-+4,
    • when x%3E=0 y+=+x+-+4. This is a line that should be easy to graph. But just graph it where x%3E=0. In other words graph y+=+x+-+4 just from the y-axis and to the right.
    • when x+%3C+0, y+=+-x+-+4. This is another line. Graph this line just where x+%3C+0. In other words graph y+=+-x+-4 just to the left of the y-axis.
    So the graph of y+=+abs%28x%29+-+4 is composed of two "half-lines", one to the right of the y-axis and one to the left.
  • Brute force. Just build a table of values and plot the points until you can see what is going on.

Domain.
There is no reason to exclude any values for x because any number can be put into an absolute value. So the domain is all Real numbers.

Range.
What are the possible vales of abs%28x%29+-+4? Of course the answer to this question will depend on the possible values of abs%28x%29. The absolute value of any number can only be positive or zero. It can never be negative. Another way to say this is that absolute values are always greater than or equal to zero. Then abs%28x%29+-+4 must be greater than or equal to -4. So the range is all Real numbers that are greater than or equal to -4.

If the logic used in calculating the range is unclear to you, here's some Algebra that may explain it more clearly:
y+=+abs%28x%29+-+4
Add 4 to both sides:
y+%2B+4+=+abs%28x%29
Since all absolute values are greater than or equal to zero and since this equation tells us that y+4 equals an absolute value:
y+%2B+4+%3E=+0
Subtract 4 from each side:
y+%3E=+-4