Determining Function Characteristics: Increasing/Decreasing and Concavity

Increasing or Decreasing Functions

• Determination Method
  • Use a table to observe whether function values increase or decrease as ( x ) increases.
• Examples
  • First Table:
    • As ( x ) increases, function values decrease, indicating a decreasing function.
  • Second Table:
    • Similar to the first, function values decrease, indicating a decreasing function.

Concavity: Concave Up or Concave Down

• Determination Method
  • Evaluate if the rate of change increases or decreases as ( x ) increases.
  • Calculate the average rate of change by dividing the change in function values by the change in ( x ) values.
• Concavity Identification
  • Concave Up: Rate of change increases as ( x ) increases.
  • Concave Down: Rate of change decreases as ( x ) increases.

Examples and Calculations

• First Table
  • Average rates of change: -17, -12, -7, -6, -4.
  • Values increase (less negative), indicating the function is concave up.
  • Conclusion: Decreasing and concave up.
• Second Table
  • Average rates of change: -6, -11, -18, -23, -25.
  • Values decrease (more negative), indicating the function is concave down.
  • Conclusion: Decreasing and concave down.

Graphical Verification

• First Graph
  • Down-hill slope: Decreasing.
  • Upward facing cup shape: Concave up.
• Second Graph
  • Down-hill slope: Decreasing.
  • Downward facing cup shape: Concave down.

Additional Examples

• First Additional Table
  • As ( x ) increases, function values increase, indicating an increasing function.
• Second Additional Table
  • As ( x ) increases, function values increase, indicating an increasing function.

Additional Rate of Change Calculations

• First Additional Table
  • Average rates of change: 7, 12, 18, 25, 32.
  • Values increase, indicating the function is concave up.
  • Conclusion: Increasing and concave up.
• Second Additional Table
  • Average rates of change: 43, 35, 25, 15, 6.
  • Values decrease, indicating the function is concave down.
  • Conclusion: Increasing and concave down.

Graphic Verification

• First Additional Graph
  • Up-hill slope: Increasing.
  • Upward facing cup shape: Concave up.
• Second Additional Graph
  • Up-hill slope: Increasing.
  • Downward facing cup shape: Concave down.

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• Conclusion
  • Tables and graphs are a useful way to determine if a function is increasing or decreasing and whether it is concave up or concave down.
  • Key is to analyze the direction of the slope and the curvature formed by the points.