1. Shifting of Graphs Transformation

2. Example 1 y = f( x ) + kUp k units y = f ( x ) - kDown k units Vertical Shifting Below is the graph of a function y = f ( x ). Sketch the graphs of a)y = f ( x ) + 1 b) y = f ( x ) - 2 y = f(x) + 1 y = f (x) y = f (x)-2

3. Horizontal Shifting y = f( x + h )Left h units y = f ( x - h )Right h units Continued … Example 2. Given the graph of a function y = f ( x ). Sketch the graphs of a)y = f ( x + 3 ) b) y = f ( x – 4 ) y = f ( x ) y = f ( x + 3) y = f ( x - 4) Horizontal Shift 3 units to the left Horizontal Shift 4 units to the right

4. Continued … Can you tell the effects on the graph of y = f ( x ) y = f( x + h ) + k y = f( x + h ) - k y = f( x - h ) + k y = f( x - h ) - k Example 3 Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1 y = f( x ) y = f( x + 2 ) - 1 Left h units and Up k units Left h units and Down k units Right h units and Up k units Right h units and Down k units

5. Continued… Example 4 Below is the graph of a function. Sketch the graph of y = f( x ) y =f(x-2)-3 The graph of the absolute value is shifted 2 units to the right and 3 units down Solution:

6. Example 5 If the point P is on the graph of a function f. Find the corresponding point on the graph of the given function. 1)P ( 0, 5 ) y = f( x + 2 ) – 1 2)P ( 3, -1 ) y = 2f(x) +4 3) P( -2,4)y = (1/2) f( x-3) + 3 Solution: 1)P ( 0,5). y = f( x + 2 ) – 1 shifts x two units to the left and shifts y one unit down. The new x =0 – 2 = -2, and the new y = 5 – 1 = 4. The corresponding point is ( -2, 4 ). 2) P(3,-1). y = 2f(x) +4 has no effect on x. But it doubles the value of y and shifts it 4 units vertically up. Therefore the new x = 3(same as before ),), and the new value of y = 2 (-1 ) + 4 = 2. Therefore, the corresponding point is ( 3,2 ). 3)P(-2, 4 ). y = (1/2) f( x-3) + 3 shifts x 3 units to the right and splits the value of y in half and then shifts it 3 units up. That is, the new value of y = (1/2)(4) + 3 = 5. Therefore, the corresponding point is ( 1, 5 ).

7. Reflecting a graph through the x-axis y = -f( x)Reflection through the x-axis (x-axis acts as a plane mirror) Example 11 Note: For any point P(x,y) on the graph of y = f(x), The graph of y = - f(x) does not effect the value of x, but changes the value of y into - y Below is the graph of a function y = x2 x2. Sketch the graph of 1.y = - x2x2 xy = x 2 y = -x 2 24-4 11 000 1 -24-4 Example 6

8. Vertical Stretching y = cf( x) ( c> 1 )Vertical Stretch by a factor c y = (1/c)f ( x) ( c > 1 )Vertical Compress by a factor 1/c Note1 :When c > 1. Then 0 < 1/c < 1 Note 2 : c effects the value of y only. Example 7 Below is the graph of a function y = x2 x2. Sketch the graphs of 1.y = 5 x2x2 2. y = (1/5)x 2 xy = x 2 y=5x 2 y=1/5x 2 210.4 115.2 0000 15.2 210.4

9. Sketching a piece-wise function Example 8 Definition: Piece-wise function is a function that can be described in more than one expression. Sketch the graph of the function f if Solution: Graph y = 2x + 5 and take only the portion to the left of the line x = The point (-1, 3 ) is included. Graph y = x2 x2 and take only the portion where –1 < x < 1. Note: the points ( -1,1) and ( 1, 1 ) are not included Graph y = 2 and take only the portion to the right of x = 1. Note: y = 2 represents a horizontal line. The point (1, 2 ) is included.

10. Sketching the graph of an equation containing an absolute value Example 9 Sketch the graph of y = g ( x ) = Note: To sketch an absolute value function. We have to remember that And hence, the graph is always above the x-axis. The part of the graph that is below the x-axis will be reflected above the x-axis. Strategy: 1. Graph y = f(x) = x2.x2. Solution: 2. Graph y = f( x ) - 9 = x 2 – 9 by shifting the graph of f 9 units down 3. Graph g(x) by keeping the portion of the graph y = f( x ) - 9 = x 2 – 9 which is above the x-axis the same, and reflecting the portion where y < 0 with respect to the x-axis. 4. Delete the unwanted portion

11. Example 10 Below is the graph of y = f(x). Graph Let the animation talk about itself Solution: A picture can replace 1000 words

12. The End