Section 3.4: Graphs and Transformations
In this section we will see how changes in the algebra will affect the basic function. We
will focus on Vertical shifts, horizontal shifts, expansions, contractions, reflections, and
combinations of any of these.
I.Vertical shifts
A.Consider the three graphs: f(x) = x2 ; g(x) = x2 + 5 ; and h(x) = x2 – 7
1. Look at the graphs on the calculator and note any similarities and
differences
B.How to create a vertical shift:
1. Consider the function f and let c be any positive constant
2. The graph of g(x) = f(x) + c is the graph of f shifted c units upward
3. The graph of h(x) = f(x) – c is the graph of f shifted c units downward
C. Consider the graphs f(x) = x,
g(x) = f(x) + 5 =
h(x) = f(x) – 4 =
II. Horizontal shifts
A.Consider the three graphs:
f(x) = 2x3 ;
g(x) = 2(x + 6)3
h(x) = 2(x - 3)3
1. Look at the graphs on the calculator and note any similarities and
differences
B. How to create a horizontal shift:
1. Consider the function f and let c be any positive constant
2. The graph of g(x) = f(x + c) is the graph of f shifted c units to the left
3. The graph of h(x) = f(x – c) is the graph of f shifted c units to the right
C. Consider the graphs f(x) = x2
g(x) = f(x -2) =
h(x) = f(x + 3) =
D. Combine what we have learned so far
1. Consider f(x) = x2 and
g(x) = f(x – 3) + 5 =
III. Expansions and Contractions
A. Consider the graphs: f(x) = x2-4 ; g(x) = 3f(x) =
1. Look at the graphs on the calculator and note any similarities and
differences
B. How to create an expansion or contraction:
1. Consider the function f and let c be any positive constant
2. The graph of g(x) = cf(x) is the graph of f stretched away from the
x-axis for 𝐜 > 1 by c
3. The graph of h(x) = cf(x) is the graph of f shrunk towards x-axis for
𝟎 < 𝐜 < 𝟏 by c
C. Consider the graph f(x)= (x-2)3+2
IV. Reflections
A. Consider the graphs: f(x) = .04x3 – x
g(x) = ¼f(x) =
g(x) = -f(x) =
h(x)=4f(x) =
h(x) = f(-x) =
1. Look at the graphs on the calculator and note any similarities and
differences
B. How to create an expansion or contraction:
1. Consider the function f and let c be any positive constant
2. The graph of g(x) = -f(x) is the graph of f reflected in the x-axis
3. The graph of h(x) = f(-x) is the graph of f reflected in the y-axis
C. Consider f(x) =
x 2
g(x) = -f(x) =
h(x) = f(-x) =

D. Combine what we have learned.
1. Consider f(x) = x2
and
g(x) = 2(x-3)2-1
Note: f(x) = a(bx-c)+d
a: affects y coordinate; b: affects x coordinate; c: shifts left/right; d: shifts up/down