2.2 Vertical and Horizontal Shifts of
Graphs
Quiz

Identify the basic function with a graph as below:
Vertical Shift of graphs

Discussion 1
y
f(x) = x2
f(x) = x2+1
↑ 1 unit
f(x) = x2-2
↓ 2 unit
f(x) = x2-5
↓ 5 unit
What about shift f(x) up by 10 unit?
shift f(x) down by 10 unit?
x
Vertical Shift of Graphs

Discussion 2
y
f(x) = x3
↑ 2 unit
f(x) = x3+2
↓ 3 unit
f(x) = x3-3
x
Vertical Shift of Graphs

If c>0, then the graph of y = f(x) + c is obtained by
shifting the graph of y = f(x) upward a distance of c units.
The graph of y = f(x) – c is obtained by shifting the graph
of y = f(x) downward a distance of c units.
↑
f(x) + c
↓
f(x) - c
Horizontal Shift of graphs

Discussion 1
y
f(x) = x2
f(x) = (x+1)2← 1 unit
f(x) = (x-2)2 → 2 unit
f(x) = (x-5)2 → 5 unit
What about shift f(x) left by 10 unit?
shift f(x) right by 10 unit?
x
Horizontal Shift of Graphs

Discussion 2
y
f(x) = |x|
← 2 unit
f(x) = |x + 2|
→ 3 unit
f(x) = |x - 3|
x
Horizontal Shift of Graphs

If c > 0, the graph of y = f(x + c) is obtained by shifting
the graph of y = f(x) to the left a distance of c units. The
graph of y = f(x - c) is obtained by shifting the graph of y
= f(x) to the right a distance of c units.
f(x + c) ←
→
f(x - c)
Conclusion
y
f(x) + c
↑
f(x + c) ←
f(x)
↓
f(x) + c
→
f(x - c)
x
f(x + c)
f(x - c)
f(x) - c
f(x) - c
Combinations of vertical and horizontal
shifts

Equation  write a description
y1 = |x - 4|+ 3. Describe the transformation of f(x) = |x|.
Identify the domain / range for both.
answer: shifting f(x) up by 3 units, then shift f(x) right by
4 units. ( or shift f(x) right by 4 units, then shift f(x) up by
3 units.)
Combinations of vertical and horizontal
shifts

Description  equation
Write the function that shifts y = x2 two units left and
one unit up.
answer: y1 = (x+2)2+1
Combinations of vertical and horizontal
shifts

Graph  equation
y
Write the equation for
the graph below.
Assume each grid mark
is a single unit.
Answer:
f(x) = (x-1)3-2
x
Combinations of vertical and horizontal
shifts

Equation  graph
Sketch the graph of
y = f(x) = √x-2 -1.
How does the
transformation affect
the domain and range?
y
x
Step 1: f(x) = √x
Step 2: f(x) = √x-2
Step 3: f(x) = √x-2 -1
Combinations of vertical and horizontal
shifts

Graph & symbolic transformation  new graph
Using the given graph of
f(x), sketch the graph of
f(x) +2
f(x+2)
f(x-1) - 3
y
x
Math 101 schedule changes

1) Project 1 will be a take-home project instead of an inclass group project. The project will be posted by Wednesday,
February 9, through the MyKAPInfo link. It is due in class on
Monday, February 14.
2) Exam 1 for Math 101 will be moved from Feb 15/16 to
Feb 16/17. Group A is scheduled for Wednesday, February 16
and Group B on Thursday, February 17. The hours for testing
for both days are 7:30 am - 9:00 pm. All exams are in ST 324.
3) Correspondingly, the deadline for full credit for Skills Test
#1 is moved to Tuesday, February 15.
Homework

PG. 99: 3-45(M3), 47-65(odds)

KEY: 18, 27, 49, 51

Reading: 2.3 Stretch, Shrink & Reflect