Graph of Piecewise Function H \[ \frac{f(x)=\left\{\begin{array}{ll} -x^{2}+4 & \text { if }-3 \leq z<0 \\ 4 & \text { if } 0 \leq x<3 \\ -x+5 & \text { if } x \geq 3 \end{array}\right.}{6} \]

Piecewise functions like the one you've presented can often be visualized as a series of connected lines and curves, creating a unique graph that tells a story! For your function, from -3 to 0, it’s a downward-opening parabola due to the \(-x^2 + 4\) portion. At \(x = 0\), it jumps up to a constant value of 4, extending from \(x = 0\) to \(x = 3\). Starting at \(x = 3\), it then transitions into a straight line with a negative slope, expressed as \(-x + 5\). Understanding how piecewise functions behave in different intervals is key to mastering their graphs. A common mistake is to forget to check the endpoints and whether points are included or not (indicated by open or closed brackets). Always remember to plot each section accurately, ensuring smooth transitions at the boundaries and keeping an eye on whether you need to include a point based on the inequality signs. Happy graphing!