Graphing Techniques Transformations Objectives: Graph Functions using Horizontal and Vertical Shifts Graph Functions using Compressions and Stretches Graph Functions using Reflections about the x-axis or y-axis

At this stage, if you were asked to graph any of the library of functions defined in Section 3.4, your response should be “Yes, I recognize these functions and know the general shapes of their graphs.” Sometimes we are asked to graph a function that is “almost” like one that we already know how to graph. In this section, we look at some of these functions and develop techniques for graphing them. Collectively, these techniques are referred to as transformations.

Raise the graph of by units Lower the graph of by units Vertical Shifts: If a real number k is added to the right side of a function , the graph of the new function is the graph of shifted vertically up or down Raise the graph of by units Lower the graph of by units 2. 1.

Horizontal Shifts: if the argument x of a function is replaced by , a real number, the graph of the new function is the graph of shifted horizontally left or right Shift the graph of to the left units Shift the graph of to the right units 3. 4.

Vertical Compressions and Stretches: when the right side of a function is multiplied by a positive number , the graph of the new function is obtained by multiplying each y-coordinate of by Stretch the graph of vertically if Compress the graph vertically if 5. 6.

Horizontal Compressions and Stretches: if the argument x of a function is multiplied by a positive number , the graph of the new function is obtained by multiplying each x-coordinate of by Compress the graph of horizontally if Stretch the graph of horizontally if 7. 8.

Reflections about the x-axis and y-axis Reflect the graph of about the x-axis. Multiply by -1 Reflect the graph of about the y-axis. Multiply by -1 10. 9.

Combinations 11. 12. 13.

EX: Find the function that is finally graphed after the following transformations are applied to the graph 1. Shift up 2 units 2. Reflect about the x-axis 3. Reflect about the y-axis 4. Shift left 3 units 5. Vertical stretched by 5