Vertical and Horizontal Shifts of Graphs

1. Vertical and Horizontal Shifts of Graphs

2. Identify the basic function with a graph as below:

3. Vertical Shift of graphs • Discussion 1 y 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?

4. Vertical Shift of Graphs • Discussion 2 y f(x) = x3 f(x) = x3+2 ↑ 2 unit f(x) = x3-3 ↓ 3 unit x

5. Vertical Shift of Graphs • 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

6. Horizontal Shift of graphs • Discussion 1 y 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?

7. Horizontal Shift of Graphs • Discussion 2 y f(x) = |x| f(x) = |x + 2| ← 2 unit f(x) = |x - 3| → 3 unit x

8. Horizontal Shift of Graphs • 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) ←

9. 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.

10. 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

11. Combinations of vertical and horizontal shifts • Graph  equation Write the equation for the graph below. Assume each grid mark is a single unit. Answer: f(x) = (x-1)3-2 y x

12. 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