SECTION 2.5 Graphing Techniques: Transformations 115 Based on the Exploration, we have the following result: In Words For ( ) = > y af x a , 0, the factor a is “outside” the function, so it affects the y -coordinates. Multiply each y -coordinate on the graph of y f x( ) = by a . 2 Graph Functions Using Compressions and Stretches Exploration On the same screen, graph each of the following functions: = = = Y x Y x Y x , 2 , 1 2 1 2 3 Then create a table of values and compare the y -coordinates for any given x -coordinate. Result Figure 58 illustrates the graphs and a table of values using Desmos. Looking at the table, notice that the values for Y2 are two times the values of Y1 for each x -value. This means that the graph of = Y x 2 2 can be obtained from the graph of Y x 1 = by multiplying each y -coordinate of Y x 1 = by 2. Therefore, the graph of Y2 will be the graph of Y1 vertically stretched by a factor of 2. The values of Y3 are half the values of Y1 for each x -value. So, the graph of Y x 1 2 3 = can be obtained from the graph of Y x 1 = by multiplying each y -coordinate by 1 2 . Therefore, the graph of Y3 will be the graph of Y1 vertically compressed by a factor of 1 2 . Figure 58 Vertical stretch or compression Vertical Compression or Stretch If a function ( ) = y f x is multiplied by a positive number a , then the graph of the new function ( ) = y a f x is obtained by multiplying each y -coordinate of the graph of ( ) = y f x by a . • If a 0 1, < < a vertical compression by a factor of a results. • If a 1 > , a vertical stretch by a factor of a results. Now Work PROBLEM 49 What happens if the argument x of a function y f x( ) = is multiplied by a positive number a , creating a new function y f ax ? ( ) = To find the answer, look at the following Exploration. Exploration On the same screen, graph each of the following functions: ( ) ( ) ( ) = = = = = = = Y f x x Y f x x Y f x x x , 2 2 , 1 2 1 2 2 1 2 3 Create a table of values to explore the relation between the x - and y -coordinates of each function. (continued)
RkJQdWJsaXNoZXIy NjM5ODQ=