# 2.2 Vertical and Horizontal Shifts of Graphs

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2.2 Vertical and Horizontal Shifts of Graphs

Quiz Identify the basic function with a graph as below:

Vertical Shift of graphs
y Discussion 1 f(x) = x2 f(x) = x2+1 ↑ 1 unit f(x) = x2-2 ↓ 2 unit x f(x) = x2-5 ↓ 5 unit What about shift f(x) up by 10 unit? shift f(x) down by 10 unit?

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
y Discussion 1 f(x) = x2 f(x) = (x+1)2 ← 1 unit f(x) = (x-2)2 → 2 unit x f(x) = (x-5)2 → 5 unit What about shift f(x) left by 10 unit? shift f(x) right by 10 unit?

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 ↑ f(x) + c → f(x - c) ↓ f(x) - c f(x + c) ← y f(x) + c f(x)

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
y Graph  equation 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
y Equation  graph Sketch the graph of y = f(x) = √x-2 -1. How does the transformation affect the domain and range? 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 in- class group project.  The project will be posted by Wednesday, February 9, through the MyKAPInfo link.  It is due in class on Monday, February )  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 )  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