1. The problem is to analyze and graph the piecewise function: $$Y = \begin{cases} -2x^2 + x - 5 & \text{if } x < -2 \\ |x - 4| & \text{if } x \geq 2 \end{cases}$$ 2. First, solve for the roots of the quadratic part $-2x^2 + x - 5 = 0$ to find intercepts. 3. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -2$, $b = 1$, $c = -5$. 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = 1^2 - 4(-2)(-5) = 1 - 40 = -39$$ 5. Since $\Delta < 0$, there are no real roots; the parabola does not cross the x-axis. 6. For $x \geq 2$, the function is $y = |x - 4|$, which is a V-shaped graph with vertex at $(4,0)$. 7. The graph has two parts: - For $x < -2$, a downward opening parabola $-2x^2 + x - 5$ with vertex and no x-intercepts. - For $x \geq 2$, the absolute value function $|x - 4|$. 8. The graph is undefined between $-2$ and $2$. Final answer: The piecewise function consists of a parabola on the left side and an absolute value function on the right side, with no x-intercepts for the parabola and a vertex at $(4,0)$ for the absolute value part.