Final answer:
The function y = -4 |x+5| has a vertex at (-5, 0), zeros at x = -5, and a y-intercept at (0, -20). Its domain is all real numbers, the range is all real numbers less than or equal to 0, with end behavior as y approaching negative infinity as x approaches both positive and negative infinity. The function is constantly decreasing.
Step-by-step explanation:
When evaluating the function y = -4 |x+5|, we can find several characteristics:
- The vertex of the absolute value function is located at the point where the inside of the absolute value is zero. In this case, it is when x = -5. Thus, the vertex is (-5, 0).
- The zeros of the function are also at x = -5 because this is where the output y equals 0.
- The y-intercept occurs where x = 0. Substituting into the function gives y = -4 |0+5| = -20, so the y-intercept is at (0, -20).
- The domain of any absolute value function is all real numbers, meaning any x-value is acceptable.
- The range of this function, due to the multiplication by -4, is all real numbers less than or equal to 0. This is because the absolute value function is always non-negative, and multiplying by -4 will always give a non-positive result.
- The end behavior is that as x approaches infinity, y approaches negative infinity; and as x approaches negative infinity, y also approaches negative infinity.
- Examining the intervals, the function is decreasing the entire time since it’s being multiplied by -4.