8 f(8) = 0 (8,0) 4 f(4) = 4 (4, 4) 2 f(2) = 3 (2, 3) 6 f(6) = 3 (6, 3) Outputs. Inputs

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1 In the previous set of notes we covered how to transform a graph by stretching or compressing it vertically. In this lesson we will focus on stretching or compressing a graph horizontally, which like the horizontal shifts that were covered earlier are not as straightforward as the vertical transformations. Eample 1: Given below is a table of inputs, outputs, and ordered pairs for a function f, as well as its graph. Use this information to answer the following parts: Ordered Pairs f() (, f()) 8 f( 8) = 0 ( 8, 0) 6 f( 6) = 3 ( 6, 3) 4 f( 4) = 4 ( 4,4) f( ) = 3 (, 3) 0 f(0) = 0 (0,0) f() = 3 (, 3) 4 f(4) = 4 (4, 4) 6 f(6) = 3 (6, 3) 8 f(8) = 0 (8,0) f()

2 a. Given a new function p(), such that p() = f(), determine whether the changes to the original function will be taking place within the function (changing the inputs) or outside the function (changing the outputs)? When the changes to a function takes place inside the parentheses, such as f( ), the change occurs with the inputs, and the result is a horizontal transformation. Also, as we saw with the horizontal shifts in a previous set of notes, we do the opposite of the operation that is listed. So in this case, the function p is taking the inputs of the function f and not multiplying them by, but rather dividing them by. Taking the inputs from some original function and dividing them by results in a horizontal compression (the new inputs will be half the original inputs, so they will be squeezed closer together). So the graph of p will have inputs that are half the outputs of f, and as a result the graph of p will look narrower than the graph of f. One way to see this is by transforming the ordered pairs from the original input/output table for f. b. Complete the Transformed Table below to find the inputs and outputs of the function p, and then sketch its graph. shifts, we need to do the opposite of what is inside the Original Table f() Transformed Table NEW p() = f() 8 = = = 4 = = = = 4 6 = = 4 0

3 Original Table f() Transformed Table NEW (we divide by because we need inputs that are half the size of the original inputs in order to produce the same outputs) p() = f() p f c. How does the new function p transform the original function f? The new function p transforms the original function f by compressing the graph horizontally by dividing the inputs of f by a factor of.

4 Eample : Given below is a table of inputs, outputs, and ordered pairs for a function f, as well as its graph. Use this information to answer the following parts: Ordered Pairs f() (, f()) 8 f( 8) = 0 ( 8, 0) 6 f( 6) = 3 ( 6, 3) 4 f( 4) = 4 ( 4,4) f( ) = 3 (, 3) 0 f(0) = 0 (0,0) f() = 3 (, 3) 4 f(4) = 4 (4, 4) 6 f(6) = 3 (6, 3) 8 f(8) = 0 (8,0) f() a. Given a new function q(), such that q() = f ( 1 ), will n() be transforming the inputs or outputs of the original function f? When the changes to a function takes place inside the parentheses, such as f ( 1 ), the change occurs with the inputs, and the result is a horizontal transformation. Also, as we saw in the previous eample, we do the opposite of the operation that is listed. So in this case, the function q is taking the inputs of the function f and not multiplying them by 1, but rather dividing them by 1, which is the same as multiplying by. Taking the inputs from some original function and multiplying them by results in a horizontal stretch (the new inputs will be twice the original inputs, so they will be stretched out farther apart). So the graph of q will have inputs that are twice the outputs of f, and as a result the graph of q will look wider than the graph of f. One way to see this is by transforming the ordered pairs from the original input/output table for f. Input s

5 Original Table f() Transformed Table NEW (we multiply by because we need inputs that are twice the size of the original inputs in order to produce the same outputs) n() = 1 f() ( 8) = 16 0 ( 6) = 1 3 ( 4) = 8 4 ( ) = 4 3 Once again we transform the inputs of f by doing the opposite operation (dividing by 1 rather than multiplying by 1 ). This time the inputs of the new function q are twice the inputs of the original function f. The new function q transforms the original function f by stretching it horizontally by a factor of. (0) = () = 4 3 (4) = 8 4 (6) = 1 3 (8) = 16 0 f n

6 Remember that when changes take place INside the parentheses, those changes only effect the INputs, and we do the INverse operation. Eample 1 showed the graph of p() = f(), which was the graph of f compressed horizontally by dividing the inputs of f by a factor of. This is because the input requires us to use new -values that are half of the original -values in order to produce the same function values (outputs); this is why the graph is compressed instead stretched. f() = 4 when = 4; f() = 4 when = In this case, the points on the original graph are being squeezed toward the y-ais, but the y-intercept remain unchanged because the input of the yintercept (the -value) is 0. Eample showed the graph of q() = f ( 1 ), which was the graph of f stretched horizontally by a factor of (dividing the inputs of f by 1 produces the same result as multiplying the inputs by ). This is because the input ( 1 ) requires us to use new -values that are twice as large as the original -values in order to produce the same function values (outputs); this is why the graph is stretched instead compressed. f() = 4 when = 4; f ( 1 ) = 4 when = = ( 4)() = 8 When the change takes place outside the parentheses, do eactly what you see to the outputs. When changes take place INside the parentheses, those changes only effect the INputs, and we do the INverse operation.