1. Graphical Transformations!!!
Sec. 1.5a is amazing!!!
Homework: p odd

2. 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)

3. New Definitions 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

4. 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

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

6. 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)

7. 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)

8. 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…

9. On to vertical and horizontal stretches and shrinks…

10. Stretches and Shrinks Horizontal Stretches or 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):
Horizontal Stretches or Shrinks
a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1
Vertical Stretches or Shrinks
a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1

11. Let C be the curve defined by y = f(x) = x – 16x. Find 3
Let C be the curve defined by y = f(x) = x – 16x. Find
equations for the following non-rigid transformations of C : 1 1 1
1. C is a vertical stretch of C by a factor of 3 2 1
2. C is a horizontal shrink of C by a factor of 1/2 3 1
Let’s verify our algebraic work graphically…

12. 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

13. 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