Question

I have the question y=-|x^2 -9|
I am asked for the piecewise function. I have written
y={-x^2 +9 if x less than equal to -3}
{ x^2 -9 if x is greater than equal to 3}
I am wondering if I have my are signs backwards.My reasons for choosing the signs I have is if I plug a negative number into the first one and square it, it will be positive and then the negative out front will make it negative. Is there a trick to getting the signs right?

Answer 1

let's look at the graph.
http://www.wolframalpha.com/input/?i=plot+y%3D-%7Cx%5E2+-9%7C+%2C+x+from+-10+to+10

notice that the entire graph must lie below the x-axis, since we have y = -(some positive stuff)
so for x<-3 OR x>+3, we have y = -x^2 + 9
for -3 ≤ x ≤ 3 we have y = x^2 - 9

I usually isolate the absolute value expression,
|x^2 - 9| = -y
to get
x^2 - 9 = -y OR -x^2 + 9 = -y
y = -x^2 + 9 OR y = x^2 - 9
I then look at the critical values, which are -3 , +3
Consider a value between -3 and 3, say x = 0
then y = -9 , from y = x^2 - 9
and y = +9 , from y = -x^2 + 9

so y = x^2 - 9 must match -3 ≤ x ≤ +3
and y = -x^2 + 9 must match -3 < x OR x > 3

making a sketch is a safe way to get the signs right.