Final answer:
The equation y= -|x+4|-2 can be expressed as a piecewise function by considering two cases for the expression inside the absolute value: when x+4 is non-negative and when it is negative, resulting in two different linear equations for the segments of the graph.
Step-by-step explanation:
To express the equation y= -|x+4|-2 as a piecewise function, we recognize that the absolute value function creates two cases based on the term inside the absolute value. The piecewise function is defined by considering the expression inside the absolute value to be either non-negative or negative. Here are the steps:
- If x+4 ≥ 0, then |x+4| = x+4 and the equation becomes y= -(x+4) - 2.
- If x+4 < 0, then |x+4| = -(x+4) and the equation becomes y= -(-(x+4)) - 2 which simplifies to y= x+4 - 2.
The piecewise function can then be written as:
- For x ≥ -4: y = -x - 6
- For x < -4: y = x + 2
By understanding the behavior of the absolute value function, we can sketch the graph of this piecewise function by plotting specific (x,y) data pairs and creating separate line segments for each case.