Question 216470
There are a couple of ways we can graph this.<br>
Solution 1: If you know what the graph of {{{y = abs(x)}}} then you can figure out {{{y = -abs(x) + 1}}}:<ul><li>The "-" in front of the absolute value causes the graph to be reflected in the x-axis.</li><li>The "+1" causes the graph to be raised up 1 unit.</li><li>Putting these together, the graph of {{{y = -abs(x) + 1}}} is the graph of {{{y = abs(x)}}} reflected in the x-axis and then raised up 1.</li></ul>
Solution 2: Use what you know about absolute value to rewrite {{{y = -abs(x) + 1}}} as a piecewise function. (This will make more sense when you see it.) What do we know about absolute value? We know that<ul><li>The absolute value of any positive number is itself.</li><li>The absolute value of zero is itself</li><li>The absolute value of any negative number is the negative of its negative self (IOW, the "positive version" of itself)</li></ul>
To put this in algebraic terms:
If {{{x >= 0}}} then {{{abs(x) = x}}}
If {{{x < 0}}} then {{{abs(x) = -x}}}
Now we can use these to rewrite {{{y = -abs(x) + 1}}} in two ways, one for {{{x >= 0}}} and another for {{{x < 0}}}:
If {{{x >= 0}}} then {{{y = -x + 1}}}
If {{{x < 0}}} then {{{y = -(-x) + 1 = x + 1}}}
And we can use these to graph {{{y = -abs(x) + 1}}}. On a graph {{{x >= 0}}} is on the y-axis and to the right of the y-axis and {{{x < 0}}} is to the left of the y-axis. So we graph {{{y = -x + 1}}}, which is a simple line (slope = -1, y-intercept = 1) <i>just from the y-axis and to the right</i>. And then, on the same graph, graph {{{y = x + 1}}}, which is also a simple line (slope = 1, y-intercept of 1), <i>just to the left of the y-axis</i>.<br>
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).