Algebra 2: Graph Transformations

Graph Transformations

• Vertical Shift Up:

  • Formula: ( y = f(x) + c )
  • Shifts the graph up by ( c ) units when a constant is added at the end of the function.

• Vertical Shift Down:

  • Formula: ( y = f(x) - c )
  • Shifts the graph down by ( c ) units when a constant is subtracted from the function.

• Horizontal Shift Left:

  • Formula: ( y = f(x + c) )
  • Shifts the graph left by ( c ) units when the constant is grouped with ( x ) in addition.

• Horizontal Shift Right:

  • Formula: ( y = f(x - c) )
  • Shifts the graph right by ( c ) units when the constant is subtracted from ( x ).

• Reflection Across the X-Axis:

  • Formula: ( y = -f(x) )
  • Flips the entire graph upside down by multiplying the function by -1.

• Reflection Across the Y-Axis:

  • Formula: ( y = f(-1x) )
  • Flips the graph horizontally by replacing ( x ) with (-1x).

• Vertical Stretch:

  • Formula: ( y = cf(x) ) where ( c > 1 )
  • Stretches the graph vertically by a factor of ( c ), making it taller without moving the x-coordinates. The y-coordinates are multiplied by ( c ).

• Vertical Compression:

  • Formula: ( y = cf(x) ) where ( 0 < c < 1 )
  • Compresses the graph vertically by a factor of ( c ), making it shorter.

• X-Axis Unchanged:

  • During vertical transformations, the x-axis remains constant; only y-coordinates change based on the transformations applied.