document.write( "Question 216470: what is the absolute value function of y=-|x|+1 how do i graph it?
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Algebra.Com's Answer #163447 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
There are a couple of ways we can graph this.

\n" ); document.write( "Solution 1: If you know what the graph of $\"y+=+abs%28x%29\"$ then you can figure out $\"y+=+-abs%28x%29+%2B+1\"$ :

The \"-\" in front of the absolute value causes the graph to be reflected in the x-axis.
The \"+1\" causes the graph to be raised up 1 unit.
Putting these together, the graph of $\"y+=+-abs%28x%29+%2B+1\"$ is the graph of $\"y+=+abs%28x%29\"$ reflected in the x-axis and then raised up 1.

\n" ); document.write( "Solution 2: Use what you know about absolute value to rewrite $\"y+=+-abs%28x%29+%2B+1\"$ as a piecewise function. (This will make more sense when you see it.) What do we know about absolute value? We know that

The absolute value of any positive number is itself.
The absolute value of zero is itself
The absolute value of any negative number is the negative of its negative self (IOW, the \"positive version\" of itself)

\n" ); document.write( "To put this in algebraic terms:
\n" ); document.write( "If $\"x+%3E=+0\"$ then $\"abs%28x%29+=+x\"$
\n" ); document.write( "If $\"x+%3C+0\"$ then $\"abs%28x%29+=+-x\"$
\n" ); document.write( "Now we can use these to rewrite $\"y+=+-abs%28x%29+%2B+1\"$ in two ways, one for $\"x+%3E=+0\"$ and another for $\"x+%3C+0\"$ :
\n" ); document.write( "If $\"x+%3E=+0\"$ then $\"y+=+-x+%2B+1\"$
\n" ); document.write( "If $\"x+%3C+0\"$ then $\"y+=+-%28-x%29+%2B+1+=+x+%2B+1\"$
\n" ); document.write( "And we can use these to graph $\"y+=+-abs%28x%29+%2B+1\"$ . On a graph $\"x+%3E=+0\"$ is on the y-axis and to the right of the y-axis and $\"x+%3C+0\"$ is to the left of the y-axis. So we graph $\"y+=+-x+%2B+1\"$ , which is a simple line (slope = -1, y-intercept = 1) just from the y-axis and to the right. And then, on the same graph, graph $\"y+=+x+%2B+1\"$ , which is also a simple line (slope = 1, y-intercept of 1), just to the left of the y-axis.

\n" ); document.write( "Either way we should end up with a graph that looks somewhat like an upside down \"v\" with the point of the \"v\" at (0,1).
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