Class Notes: Graphing Cubic Functions Using Transformations

1. Parent Function and Its Graph

• Parent cubic function: ( y = x^3 )
  • Shape: Always increasing for all ( x )
  • Graph: Passes through the origin (0,0)
• Negative sign: ( y = -x^3 )
  • Effect: Reflects the graph over the x-axis (equivalent to reflecting over the origin)
  • Trend: Always decreasing as ( x ) increases
• Absolute value: ( y = |x^3| )
  • Effect: Right side remains the same; left side reflects over x-axis (all negative values become positive)
  • Shape: Resembles ( y = x^2 ) on the left

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2. Transformations Overview

Cubic functions can be modified and shifted:

[ y = a(x - h)^3 + k ]

• ( a ): Controls vertical stretch/compression and reflection
  • ( a > 0 ): Increasing
  • ( a < 0 ): Decreasing (reflection)
• ( h ): Horizontal shift (center at ( h, k ))
  • ( h > 0 ): Right
  • ( h < 0 ): Left
• ( k ): Vertical shift
  • ( k > 0 ): Up
  • ( k < 0 ): Down

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3. Examples of Transformations

Example 1: ( y = x^3 + 2 )

• Transformation: Shift up 2 units
• Center: (0, 2)

Example 2: ( y = x^3 - 3 )

• Transformation: Shift down 3 units
• Center: (0, -3)

Example 3: ( y = (x - 2)^3 )

• Transformation: Shift right 2 units
• Center: (2, 0)

Example 4: ( y = (x + 4)^3 )

• Transformation: Shift left 4 units
• Center: (-4, 0)

Example 5: ( y = -(x - 3)^3 )

• Transformation: Shift right 3 units and reflect over the x-axis
• Center: (3, 0)

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4. More Complex Example: ( y = (x - 3)^3 + 2 )

• General form: ( y = a(x-h)^3 + k )
  • ( a = 1 ): Increasing
  • ( h = 3 ): Right 3 units
  • ( k = 2 ): Up 2 units
• Center: (3, 2)
• To plot the graph accurately:
  • Make a data table centered at (3, 2)
  • Use points:
    1. ( x = 1, y = -6 )
    2. ( x = 2, y = 1 )
    3. ( x = 3, y = 2 ) (center)
    4. ( x = 4, y = 3 )
    5. ( x = 5, y = 10 )

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5. Another Example: ( y = -(x + 1)^3 + 4 )

• General form: ( y = a(x-h)^3 + k )
  • ( a = -1 ): Decreasing
  • ( h = -1 ): Left 1 unit
  • ( k = 4 ): Up 4 units
• Center: (-1, 4)
• Data table (centered at (-1,4)):
  • ( x = -3, y = 12 )
  • ( x = -2, y = 5 )
  • ( x = -1, y = 4 ) (center)
  • ( x = 0, y = 3 )
  • ( x = 1, y = -4 )

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6. Summary Steps for Graphing Cubic Functions with Transformations

1. Identify the center (( h, k )) from the equation.
2. Determine if the graph is increasing or decreasing (the sign of ( a )).
3. Make a data table centered at (( h, k )), using points to the left and right for an accurate sketch.
4. Plot at least five points: Center and two points on each side spaced equally, using the rule that shifts change cubically (by 1 and 8 for moves of 1 and 2 units from the center).
5. Draw the general shape: Based on the sign of ( a ) (increasing or decreasing cubic).

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7. Key Takeaways

• Cubic graphs are shaped by their transformations.
• Always locate the center and note direction (increasing or decreasing).
• Using a data table with chosen x-values (centered around the center) makes accurate graphing easier.
• Each transformation (shift, stretch, reflection) can be identified from the equation structure.

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Note: For structured learning or quick access to similar videos and examples, refer to the instructor's website.