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Section 2.5 Transformations of Functions * Graphs of Common Functions Table 2.3 (page 242 on the textbook) * Vertical Shifts Let f be a function and c a positive real number. The graph of y f (x) c is the graph of y f (x) shifted c units vertically upward. The graph of y f (x) c is the graph of y f (x) shifted c units vertically downward. Example 1) Use the graph of f (x) x to obtain the graph of g(x) x 3. * Horizontal Shifts Let f be a function and c a positive real number. The graph of y f (x c) is the graph of y f (x) shifted to the left c units. The graph of y f (x c) is the graph of y f (x) shifted to the right c units. Example 2) Use the graph of f (x) x to obtain the graph of g(x) x 4 . 1 Example 3) Use the graph of f (x) x 2 to obtain the graph of g(x) (x 2) 2 3 . *Reflections of Graphs The graph of y f (x) is the graph of y f (x) reflected about the x-axis. Example 4) Use the graph of f (x) x to obtain the graph of g(x) x . The graph of y f (x) is the graph of y f (x) reflected about the y-axis. Example 5) Use the graph of f (x) 3 x to obtain the graph of g(x) 3 x . 2 * Vertical Stretching and Shrinking Let f be a function and c a positive real number. If c 1, the graph of y cf (x) is the graph of y f (x) vertically stretched by multiplying each of its y-coordinates by c. If 0 c 1, the graph of y cf (x) is the graph of y f (x) vertically shrunk by multiplying each of its y-coordinates by c. 1 3 3 Example 6) Use the graph of f (x) x to obtain the graph of g(x) 2 x . * Horizontal Stretching and Shrinking Let f be a function and c a positive real number. If c 1, the graph of y f (cx) is the graph of y f (x) horizontally shrunk by dividing each of its x-coordinates by c. If 0 c 1, the graph of y f (cx) is the graph of y f (x) horizontally stretched by dividing each of its y-coordinates by c. Example 7) Use the graph of f (x) x to obtain the graphs of g(x) 2x and 1 h(x) x . 2 3 * Sequences of Transformations (HSRV) Let f (x) be a function and c be a positive real number. To Graph: Draw the Graph of f and: Horizontal shifts Stretching Vertical or Shrinking Horizontal Reflection About the x-axis About the y-axis Vertical shifts Changes of Coordinates (each point) y f (x c) Shift c units Left. Subtract c from each xcoordinate. y f (x c) Shift c units Right. Add c from each xcoordinate. Multiply each yy cf (x), Stretch vertically by multiplying y by c . coordinate by c . c 1 Shrink vertically by Multiply each yy cf (x), multiplying y by c . coordinate by c. 0 c 1 Shrink horizontally by Divide y f (cx) , each x-coordinate dividing x by c. by c. c 1 Stretch horizontally by Divide each x-coordinate y f (cx) , dividing x by c. by c. 0 c 1 y f (x) Reflect about x-axis. Replace each ycoordinate with its opposite. y f (x) Reflect about y-axis. Replace each xcoordinate with its opposite. y f (x) c Raise by c units. Add c from each ycoordinate. y f (x) c Lower by c units. Subtract c from each ycoordinate. A function involving more than one transformation can be graphed by performing transformations inthe following order. 1. Horizontal shifting 2. Stretching or shrinking 3. Reflecting 4. Vertical shifting 4 Example 8) Use the graph of f (x) x 2 to graph g(x) 2(x 3) 2 3 . Step 1 Find the basic function unless it is given. of the changesfrom the basic function to the given function in Step 2 Find the sequence the HSRV order. x2 2(x 3) 2 3 Step 3 Describe the changes of graph of each transformation. 1. 2. 3. Step 4 Pick several points of the basic function of graph and find the transformations of each point Step 5 Draw graphs with the points. 5 Example 9) Use the graph of y f (x) to graph g(x) 1 f (2x) 3 . 2 Step 1 Find the basic function unless it is given. Step 2 Find the sequence of the changes from the basic function to the given function in the HSRV order. f (x) 1 f (2x) 3 2 Step 3 Describe the changes of graph of each transformation. 1. 2. 3. 4. Step 4 Pick several points of the basic function of graph and find the transformations of each point Step 5 Draw graphs with the points. 6