Graph of Piecewise Function f(x)

Here’s a detailed description of the piecewise function f(x) and its graph, structured to guide comprehension and sketching:

Piecewise Function Definition

The function f(x) is defined in three pieces over distinct intervals:

1. First Piece (−5≤x<2):
   • Expression: f(x)=2−x
   • Type: Linear (straight line), decreasing (slope = −1).
2. Second Piece (2≤x≤5):
   • Expression: f(x)=x−2
   • Type: Linear (straight line), increasing (slope = +1).
3. Third Piece (x<−5 or x>5):
   • Expression: f(x)=161​x2−41​x+1
   • Type: Quadratic (parabolic curve), opening upward (coefficient of x2 is positive).

Key Features to Label on the Graph

1. Intercepts:
   • y-intercept: (0,2) (where x=0, falls in the first piece: f(0)=2−0=2).
   • x-intercept: (2,0) (where f(x)=0, falls at the boundary of the first and second pieces: 2−x=0⇒x=2).
2. Critical/Boundary Points:
   • x=2: The junction point between the first and second pieces, where the function changes from decreasing to increasing.
   • x=−5 and x=5: Boundary points separating the linear pieces from the quadratic piece.
3. Behavior Annotations:
   • Decreasing Interval: [−5,2) (first piece).
   • Increasing Interval: [2,5] (second piece).
   • Non-Differentiable Points: x=−5,2,5 (sharp turns at these points due to abrupt changes in slope).
4. Quadratic Curve Details (x<−5 or x>5):
   • Vertex: Complete the square to find the vertex:f(x)=161​(x2−4x)+1=161​(x−2)2+43​
     Vertex at (2,43​), but note that this quadratic is only graphed for x<−5 or x>5, so the main branches extend outward from these intervals.
   • End Behavior: As x→±∞, the quadratic term dominates, so f(x)→+∞.

Step-by-Step Sketching Guide

1. Graph the First Piece (−5≤x<2):
   • Plot endpoints:
     • At x=−5: f(−5)=2−(−5)=7 → point (−5,7) (closed circle, included in the interval).
     • At x=2: f(2)=2−2=0 → point (2,0) (open circle for the first piece, since x<2).
   • Draw a straight line connecting these points, decreasing from left to right.
2. Graph the Second Piece (2≤x≤5):
   • Plot endpoints:
     • At x=2: f(2)=2−2=0 → point (2,0) (closed circle, included in the interval).
     • At x=5: f(5)=5−2=3 → point (5,3) (closed circle).
   • Draw a straight line connecting these points, increasing from left to right.
3. Graph the Third Piece (x<−5 or x>5):
   • For x<−5:
     • Calculate a few points (e.g., x=−6):f(−6)=161​(36)−41​(−6)+1=49​+23​+1=419​=4.75
     • The curve approaches x=−5 from the left, connecting to the first piece’s endpoint (−5,7) with a sharp turn (non-differentiable).
   • For x>5:
     • Calculate a few points (e.g., x=6):f(6)=161​(36)−41​(6)+1=49​−23​+1=47​=1.75
     • The curve approaches x=5 from the right, connecting to the second piece’s endpoint (5,3) with a sharp turn (non-differentiable).

Final Graph Annotations

• Mark all intercepts, boundary points, and critical points with labeled dots.
• Use arrows to indicate the quadratic curve extending to infinity in the x<−5 and x>5 regions.
• Add notes near x=−5,2,5 indicating "non-differentiable" or "sharp turn."
• Shade or color-code each piece for clarity, ensuring the linear segments and quadratic curve are distinct.

This visualization highlights the function’s 分段特性 (piecewise nature), continuity at boundary points, and abrupt changes in slope, providing a clear overview of its behavior across all intervals.