HPC 1.6 Day 1 – Graphical Transformations
Understanding how algebraic alterations change the shapes, sizes, positions, and orientations of graphs is
helpful for understanding the connection between algebraic and graphical models of functions. In this section
we relate graphs using transformations, which are functions that map real numbers to real numbers. By
acting on the x-coordinates and y-coordinates of points, transformations change graphs in predictable ways.
Rigid transformations, which leave the size and shape of a graph unchanged, include horizontal translations,
vertical translations, reflections, or any combination of these. Non-rigid transformations which generally
distort the shape of a graph, include horizontal or
vertical stretches and shrinks.
Vertical and Horizontal Translations:
A vertical translation of the graph of y  f  x  is
a shift of the graph up or down in the coordinate
plane. A horizontal translation is a shift of the
graph to the left or the right.
Problem #1: Give the equations of the solid line graphs below. (Dashed line is parent function y  x3 .)
1a) ____________________ 1b) ____________________ 1c) ____________________
Reflections Across the Axes:
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. The
graphs of problem #3 suggest that a reflection
across the x-axis results when y is replaced by -y,
and a reflection across the y-axis results when x is
replaced by -x.
1
Problem #2: Accurately sketch the following graphs on the coordinate planes below without a calculator.
f  x  x
g  x   x
h  x  x
Which axis are g(x) and h(x) reflected across? _______________________________________________
Vertical Stretches and Shrinks
We now investigate what happens when we
multiply all the y-coordinates of a graph by a
fixed real number. (Note: We won’t consider
horizontal stretches & shrinks until we work
with transformations of trigonometry functions.)
Problem #4: Accurately sketch the following graphs on the coordinate planes below without a calculator.
f  x   2x
g  x   3  2x
h  x 
2
1 x
2
4
Function Notation:
We use function notation to note a transformation of another “parent function.” (Often the parent function
will be one of the 12 basic functions.) The grid on the next page will help illustrate this notation.
Parent Function
Transformation(s)
f  x 
1 up
3 to the right
f  x 
x
Function Notation
g  x   f  x  3  1
1
1  e x
Equation of
Transformed
Function
y
y
x  3 1
Point Rule
P ( x, y ) 
( x  3, y  1)
1
2
1  e  ( x 5)
y   x  7  3
2
f  x    x
reflection over y-axis
4 to the left
y  2x2  3
y    x  2  3
3
f  x 
g  x   f   x  3  5
1
x
g  x    f  x  3  7
f  x  x
f  x   ln x
f  x   sin x
vertical scaling by
factor of 3
4 up
g  x   3 f  x  3  7
Problem #5: Now rewrite the functions you graphed on the prior pages in function notation…
Order of Transformations:
When a function has multiple transformations from a parent function, there are different orders that the
transformations can be applied. There are multiple approaches that will yield accurate results, and some
approaches which yield inaccurate results. In this class we will use the following priority when applying
transformations:
1. Apply reflections
2. Apply stretches & shrinks
3. Apply translations (vertical & horizontal “shifts”)
You can transform any point of the parent function following the above prioritization and find its
transformed point. Typically we will only transform three-five (3-5) key points and then sketch the graph
based on our understanding of the original graph and the transformations applied.
3