Ordered Pairs. 2 f(2) = 4 (2, 4) 1 f(1) = 1 (1, 1) Outputs. Inputs

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1 In the previous set of notes we looked at vertical shifts, where the graph of some iginal function was moved up down by adding numbers to subtracting numbers from the outputs of a function. In this set of notes we will focus on hizontal shifts, which are not as straightfward as the vertical shifts. Eample 1: Given below is a table of inputs, outputs, and dered pairs f the function f() = 2, as well as its graph. Use this infmation to answer the following parts: Ordered Pairs f() = 2 (, f()) 2 f( 2) = 4 ( 2, 4) 1 f( 1) = 1 ( 1, 1) 0 f(0) = 0 (0, 0) 1 f(1) = 1 (1, 1) 2 f(2) = 4 (2, 4) f()

2 a. How can we write the function j() = ( + 3) 2 in terms of the iginal function f() = 2? In other wds, what do we need to replace with in the iginal function f to make it ( + 3) 2? As stated in the previous lesson, the benefit to writing the new function j in terms of the iginal function f is to see whether we will be changing () = f( + 2) b. Given that j() is a transfmation of the iginal function f(), is the change taking place within the function (changing the inputs) outside the function (changing the outputs)? Since j() = f( + 2), I can see that the change is taking place inside the parentheses. The iginal function provided at the beginning of this c. Complete the Transfmed Table below to find the inputs and outputs of the function j, and then sketch its graph. Original Table f() (what are we doing to our old inputs to get our new inputs?) Transfmed Table j() = ( + 3) 2 j() = f( + 3) To find the new transfmed inputs f the function j, I need to think about what new inputs would need to be plugged in to the function j() = ( + 2) 2 in der to produce the same outputs that the function f() = 2 produced. In other wds, what -values would make ( + 2) 2 = 4? And what inputs would make ( + 2) 2 = 1, and what inputs would make ( The + 2) reason 2 = 0? j() = ( + 3) 2 shifts the graph of f() = 2 to the left instead of the right is because ( we + need 2) 2 to = use 4 smaller inputs to produce the same iginal outputs of the function f. The function j() = ( + 3) 2 is still taking those new inputs, adding 3 to them, and then squaring the sum. We are just subtracting 3 from the old inputs in der to get new inputs that produce the same iginal outputs.

3 Original Table f() produce the To find the new transfmed same outputs) inputs f the function j, I need to think about what new inputs would 5 need to be plugged 4 in to the function j() = ( + 2) 2 in der to produce 4 the same outputs 1 that the function j() (we subtract 3 because we need smaller inputs to Transfmed Table f j() = ( + 3) 2 j() = f( + 3) On eamples like this I simply think of j() = ( + 3) 2, j() = f( + 3), as indicating that I need to use smaller inputs in der to produce the eact same outputs, so I simply subtract 3 from each of the iginal inputs to get the new inputs. Another way of finding the new inputs that produce that same outputs is to take the new function and setting it equal to the outputs you want. ( + 3) 2 = = ± 4 = 3 ± 2 = 5, 1 a. How does the new function j transfm the iginal function f?

4 Eample 2: Given below is a table of inputs, outputs, and dered pairs f the function f() = 2, as well as its graph. Use this infmation to answer the following parts: Ordered Pairs f() = 2 (, f()) 2 f( 2) = 4 ( 2, 4) 1 f( 1) = 1 ( 1, 1) 0 f(0) = 0 (0, 0) 1 f(1) = 1 (1, 1) 2 f(2) = 4 (2, 4) f() a. How can we write the function k() = ( 2) 2 in terms of the iginal function f() = 2? Once again substitute f() in f 2 b. Is the new function k() transfming the inputs the outputs of the iginal function f()? c. Complete the Transfmed Table below to find the inputs and outputs of the function k, and then sketch its graph. Original Table f() (what are we doing to our old inputs to get our new inputs?) Transfmed Table k() = ( 2) 2 k() = f( 2)

5 Original Table f() (we add 2 because we need larger inputs to produce the same outputs) Transfmed Table k() = ( 2) 2 k() = f( 2) In der to show that the new inputs will be one plus the old inputs, we could go through the same process of proving this algebraically f k Keep in mind that when we do the opposite operation, it is just to find new inputs that we can plug into the new function but still produce the same outputs we had befe. If the function is k() = ( 2) 2, we are still taking the new inputs, subtracting 2 from them, and then squaring the result. So we re not changing the function, we are just trying to get new inputs so the function produces the same outputs as befe. The new function k transfms the iginal function f by shifting it to the right 2 units.

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 j() = ( + 3) 2, which is the graph of f() = 2 shifted left 3 units. j shifts f to the left 3 units because 3 is being subtracted from the inputs of f. The reason we are subtracting instead of adding is because the input + 3 requires us to use new values which are smaller than the iginal -values in der to produce the same function values (outputs); this is why the graph is shifted to the left instead of the right. f() = 0 when = 0; f( + 3) = 0 when = 3 Eample 2 showed the graph of k() = ( 2) 2, which is the graph of f() = 2 shifted right 2 units. k shifts f to the right 2 units because 2 is being added to each of the inputs of f. The reason we are adding instead of subtracting is because the input 2 requires us to use new -values which are larger than the iginal -values in der to produce the same function values (outputs); this is why the graph is shifted to the right instead of the left. f() = 0 when = 0; f( 2) = 0 when = 2 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. This will be true regardless of the operation; if a change is occurring outside the parentheses, do what you see to the outputs, if a change is occurring inside the parentheses, do the opposite of what you see to the inputs.