Graphical Transformations!!! Sec. 1.5a is amazing!!!

New Definitions Transformations – functions that map real numbers to real numbers Rigid Transformations – leave the size and shape of a graph unchanged (includes translations and reflections) Non-rigid Transformations – generally distort the shape of a graph (includes stretches and shrinks)

Rigid Transformations Vertical Translation – of the graph of y = f(x) is a shift of the graph up or down in the coordinate plane Horizontal Translation – a shift of the graph to the left or the right

Translations Let c be a positive real number. Then the following trans- formations result in translations of the graph of y = f(x): Horizontal Translations y = f(x – c) a translation to the right by c units y = f(x + c) a translation to the left by c units Vertical Translations y = f(x) + c a translation up by c units y = f(x) – c a translation down by c units

Each figure shows the graph of the original square root function, along with a translation function. Write an equation for each translation.

New Definitions Rigid Transformations Points (x, y) and (x, –y) are reflections of each other across the x-axis. Points (x, y) and (–x, y) are reflections of each other across the y-axis. (x, y) (–x, y) (x, –y)

Reflections The following transformations result in the reflections of the graph of y = f(x): Across the x-axis y = – f(x) Across the y-axis y = f(–x)

Find an equation for the reflection of the given function across each axis: Across the x-axis: Across the y-axis: Let’s support our algebraic work graphically…

On to vertical and horizontal stretches and shrinks…

Stretches and Shrinks Let c be a positive real number. Then the following trans- formations result in stretches or shrinks of the graph of y = f(x): a stretch by a factor of cif c > 1 a shrink by a factor of cif c < 1 Horizontal Stretches or Shrinks a stretch by a factor of cif c > 1 a shrink by a factor of cif c < 1 Vertical Stretches or Shrinks

Let C be the curve defined by y = f(x) = x – 16x. Find equations for the following non-rigid transformations of C : C is a vertical stretch of C by a factor of C is a horizontal shrink of C by a factor of 1/2 13 Let’s verify our algebraic work graphically…

Whiteboard problems… Describe how the graph of can be transformed to the given equation. reflect across x-axis shift right 5 reflect across y-axis shift right 3 Describe how the graph of can be transformed to the given equation. vertical stretch of 2 horiz. shrink of ½ horiz. stretch of 5 vertical shrink of 0.3

More whiteboard problems… Describe how to transform the graph of f into the graph of g. right 6 reflect across x-axis, left 4 Describe the translation of to Reflected across x-axis Vertical stretch factor of 2 Shift left 4 Shift up 1 Homework: p odd