To determine which of these equations has no x-intercepts, we need to analyze each expression to see if the graph intersects the x-axis (where y = 0).
-
−|x|−3: This graph represents a negative absolute value function shifted down by 3. The entire function will always be below 0 (y < 0) because the minimum value occurs when |x| = 0, making the function equal to -3. Therefore, this graph has no x-intercepts.
-
|x|−3: This is an absolute value function shifted down by 3. The graph intersects the x-axis at x = -3 and x = 3, so it has x-intercepts.
-
−|x|+3: This is a negative absolute value function shifted up by 3. The maximum point occurs at |x| = 0 (x = 0), which gives us y = 3. As x moves away from 0, the value of y will decrease, and it will intersect the x-axis (specifically at x = -3 and x = 3). So, this graph does have x-intercepts.
-
|x|+3: This absolute value function is always above 0, with its minimum value at 3 (when x = 0). Consequently, it also has no x-intercepts.
Thus, the equations with no x-intercepts are:
- −|x|−3
- |x|+3
However, since the question asks for a singular situation, the most clear situation where the graph has no x-intercepts is −|x|−3.