Because ends at and ends at, we can see that the - values have been compressed by, because. The graph of looks like the graph of horizontally compressed. We do the same for the other values to produce Table 14.īecause each input value has been doubled, the result is that the function has been stretched horizontally by a factor of 2.ĮXAMPLE 18 Recognizing a Horizontal Compression on a Graph Our input values to will need to be twice as large to get inputs for that we can evaluate. Notice that we do not have enough information to determine because, and we do not have a value for in our table. The formula tells us that the output values for are the same as the output values for the function at an input half the size. Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.ĮXAMPLE 17 Finding a Horizontal Stretch for a Tabular FunctionĪ function is given as Table 13. See Figure 24 for a graphical comparison of the original population and the compressed population.įigure 24 (a) Original population graph (b) Compressed population graph In other words, this new population,, will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. Set where for a compression or for a stretch.ĮXAMPLE 16 Graphing a Horizontal Compression.Write a formula to represent the function.HOW TO Given a description of a function, sketch a horizontal compression or stretch. If, then there will be combination of a horizontal stretch or compression with a horizontal reflection.If, then the graph will be stretched by. If, then the graph will be compressed by.We do the same for the other values to produce Table 11. The formula tells us that the output values of are half of the output values of with the same inputs. HOW TO Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.ĮXAMPLE 14 Finding a Vertical Compression of a Tabular FunctionĪ function is given as Table 10. The input values,, stay the same while the output values are twice as large as before. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. This means that for any input, the value of the function is twice the value of the function. Symbolically, the relationship is written as The following shows where the new points for the new graph will be located. If we choose four reference points, and we will multiply all of the outputs by 2. Sketch a graph of this population.īecause the population is always twice as large, the new population's output values are always twice the original function's output values. The graph is shown in Figure 20.Ī scientist is comparing this population to another population,, whose growth follows the same pattern, but is twice as large. If, the graph is either stretched or compressed and also reflected about the -axis.Ī function models the population of fruit flies. If, the graph is compressed by a factor of. If, the graph is stretched by a factor of.HOW TO Given a function, graph its vertical stretch. If, then there will be combination of a vertical stretch or compression with a vertical reflection.If, then the graph will be compressed.Given a function, a new function, where is a constant, is a vertical stretch or vertical compression of the function. Figure 19 shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.įigure 19 Vertical stretch and compression If the constant is greater than 1, we get a vertical stretch if the constant is between 0 and 1, we get a vertical compression. When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. Each change has a specific effect that can be seen graphically. We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. We now explore the effects of multiplying the inputs or outputs by some quantity. Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph.
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