Lesson 2.2.8: Linear Functions in Transformations

Objectives

• Graph linear equations using transformations.
• Identify transformations in linear equations.

Parent Function

• Equation: ( y = x )
• Characteristics:
  • Y-intercept: (0,0) (since ( b = 0 ))
  • Slope: 1 (since there is no coefficient in front of ( x ))
  • Graph: Passes through origin with a slope of 1 (up 1, over 1)
  • Constant Rate of Change: Slope remains constant across the graph

Transformations in Linear Functions

• Slope (a):
  • Change in slope is referred to as "a"
  • Greater than 1: "Stretch" (line becomes steeper)
  • Less than 1: "Shrink" (line becomes less steep)
• Y-intercept (b):
  • Positive b: Shift up
  • Negative b: Shift down
• Reflection:
  • Negative sign in front of a: Reflection over the y-axis

Examples

• Equation: ( y = x + 5 )

  • Transformation: Shift up 5
  • Graph: Line shifts up by 5 units

• Equation: ( y = x - 3 )

  • Transformation: Shift down 3
  • Graph: Line shifts down by 3 units

• Equation: ( y = 4x )

  • Transformation: Stretch by 4
  • Graph: Line becomes steeper, slope = 4

• Equation: ( y = \frac{1}{3}x )

  • Transformation: Shrink by ( \frac{1}{3} )
  • Graph: Line becomes less steep

• Equation: ( y = -x )

  • Transformation: Reflection
  • Graph: Line reflects over y-axis

Combining Transformations

• Equation: ( y = -5x )

  • Transformations: Reflection and stretch of 5
  • Graph: Reflected and steeper

• Equation: ( y = 2x + 4 )

  • Transformations: Stretch of 2, shift up 4
  • Graph: Less steep and shifted up

• Equation: ( y = 3x - 6 )

  • Transformations: Stretch of 3, shift down 6
  • Graph: Increased steepness and shifted down

Practice

• Given a linear equation, identify transformations and graph the equation:

  • Example: ( y = x - 8 )
    • Transformation: Shift down 8
    • Graph: Plot y-intercept at (0, -8), slope is 1
  • Example: ( y = 4x + 2 )
    • Transformations: Stretch of 4, shift up 2
    • Graph: Plot y-intercept at (0, 2), slope is 4

• Example: ( y = -\frac{1}{2}x + 5 )

  • Transformations: Reflection, shrink of ( \frac{1}{2} ), shift up 5
  • Graph: Plot y-intercept at (0, 5), slope leads down

Conclusion

• Types of Transformations:

  • Reflections: Due to negative signs
  • Stretches/Shrinks: Affects slope
  • Shifts: Affects y-intercept

• Understanding transformations aids in graphing more complex functions in the future.