Ch. 1 – Function Transformations
Basic Functions/Horizontal & Vertical Translations
Here are the Basic Functions (and their coordinates!) you need to get familiar with.
1. Quadratic functions (a.k.a. parabolas)
y  x2
Ex. y  ( x  2)2  1
2. Radical functions (a.k.a. square root function)
y x
Eg1.
Ex. y  x  3  4
Find the equations for the base functions and their transformed graphs.
a)
Base function:
Transformed function:
b)
Base function:
Transformed function:
Eg3.
Transform the following graph. Describe the transformations in words.
Given:
y  f ( x)
Graph:
y  f ( x  2)  3
Transformation: ________________________________________________________________
Regardless of the type of function y  f ( x) , the transformed function y  f ( x  h)  k means:
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Vertical & Horizontal Reflections
A reflection can be identified with a “negative sign.” A reflection is a mirror image of a given
function. Let’s explore the effect of having a “negative sign” at different locations of a function –
use a graphing calculator to check!
Eg1.
Graph y  x 2 and y   x 2 on the same grid.
Eg2.
Graph y  x and y   x on the same grid.
Observations:
*
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------*
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Expansions and Compressions
Eg1. Consider the following function:
f  x  x
Graph the indicated function using the table of values on the graphing calculator:
a) y  2 f  x 
b) y 
1
f  x
2
c) y  f  2 x 
1 
d) y  f  x 
2 
Combining Transformations
Does order matter? Let’s explore…
Eg1.
Graph
y x
x
y
Vertical Expansion vs. Vertical Translation:
1. VE by factor of 2
2. VT by 1 unit down
(Reverse Order)

1. VT by 1 unit down
2. VE by a factor of 2
CREATING A WRITTEN FUNCTION [y=f(x)]: Follow the instructions in the given order.
a) VE of 2
b) HE of 2
VT down 3
VT up 1
HT right 1
VC of 1/6
HC of 1/3
HT left 6
Eg3.
Describe the order of transformations that occur for the following functions.
a) y  3 f  2x   4
b) y  f  2  x  1 
c) y  f (2 x  4)
1

d) y  2 f   x  1   5
3

Eg4.
Given y  f ( x) on the right.
a) Describe the transformations below.
y  2 f 1  2 x   3
b) Graph the indicated function transformations on the grid provided.
** With regards to order, any vertical change is NOT in competition with any horizontal change!