Topic 2 Transformations of Functions

Transcription

1 Week Topic Transformations of Functions Week Topic Transformations of Functions This topic can be a little trick, especiall when one problem has several transformations. We re going to work through each tpe of transformation in order, and then do a few eamples putting them all together. As usual, make sure ou have the basics down well or the more complicated eamples will just be too much at once. This is in the tetbook on pages It s not the high point of the book, but this subject can prove difficult, so I suggest reading these notes (and taking our own notes), and then reading the book and supplementing our notes with additional insights. The book s eamples and are useful to demonstrate how to break down a transformed function into the basic steps. The book makes it look eas, but in practice ou ll probabl have to tr a few times before getting it right. Shifts Vertical Shifts The idea here is simple. We take a graph and shift it up or down on the coordinate aes. This is a modification to the values. For eample, if we want to move a graph up units, we have to add to each value. Let s take the square function f ( ). We can make a table of values for this function. Table The Square Function and a Vertical Shift values values New values after shifting up b

2 Week Topic Transformations of Functions It s simple to confirm that the formula gives the new values. In function notation, we can write a new name, like f ( ) ( ) new, or since we ve created a new function, we could give it f, or we could call it g or h or whatever ou like. Figure shows both the original graph and the transformed function. Figure The Squared Function and a Transformation of it f()=^ f()+=^ As another eample, let s shift g ( ) down b 0 units. That means subtracting 0 from all the values, so we compute g ( ) 0 0. Figure shows a graph of this function before and after this transformation. Figure The Absolute Value Function and a Transformation of it -g()=abs() -g()-0=abs() Vertical Shift Summar To shift a function f ( ) up b c units, compute f ( ) c To shift a function f ( ) down b c units, compute f ( ) c

3 Week Topic Transformations of Functions Horizontal Shifts Shifting a function horizontall requires affecting the values. This is a little more complicated than the vertical shifts, so let s look at an eample. Recall the function f ( ) has a domain of [0, ) because we can square root 0, but we can t square root an negative numbers. Figure shows a table of values and graph for this function. Figure A Table and Graph for the Square Root Function f()=sqrt() 0 0 Let s compare this to the transformed function g ( ) 0. This is an inside transformation adding a constant, so it should provide a horizontal shift like adding the constant to the outside creates a vertical shift. This new function g will take the input ( value), add 0 to it, and then square root that results. Figure shows a table of values and a graph for the transformed function g. Figure A Table and Graph of the Transformed Square Root Function g()=sqrt(+0) -0-0 Replacing with + 0 adds 0 to the values, but this results in a shift to the left. To understand wh, look at order of operations. After ou add 0, ou re square rooting a larger number, which results in a larger value. Hence, notice from the table that smaller values result in the same values as the original function f. Also notice, the domain of g ( ) is [ 0, ), so that terminal end on the function graph shifts ten units left itself. The rest of the graph shifts likewise.

4 Week Topic Transformations of Functions Let s look at another function, h ( ). Figure shows a graph of this function. Figure A Graph of h() h()=/(+^)) Let s also compute h ( ) b replacing ever in the formula for h with ( + ): h( ) h ( ) In the new formula, we add to and then do the rest of the operations. For eample, h(0), but if we put = 0 into h ( ), we get h (). However, if we put = into h ( ), we get h (0). So an value of 0 in the original function is the same as an value of in the transformed function. In other words, (0, ) is on the graph of the original function, and (, ) is on the graph of the transformed function. Figure shows a graph of the transformed function. Figure A Graph of h(+) h(+)=/(+(+)^)) Horizontal Shift Summar To shift a function f ( ) right b c units, compute f ( c) To shift a function f ( ) left b c units, compute f ( c)

5 Week Topic Transformations of Functions Shift Eample Problems. Complete the table b writing the formula for the function designated. Shifts f ( ) right units f( ) f ( ) f ( ) log( ) down units up 7 units left units. Identif the basic function f ( ) and describe the shift in words. Functions Basic Function Shift g ( ) f ( ) Shifted units down g ( ) g ( ) g ( ) g ( ) g ( ) g ( ) Check answers on our graphing utilit. It s crucial ou have a wa to graph the transformations as ou go.

6 Week Topic Transformations of Functions Reflections ais reflections Reflecting a graph around the ais is flipping it over the ais. Figure 7 shows an eample of a function s graph and after it s transformed b an ais reflection. f()=(^--)/ points on the graph Figure 7 A Function and Its ais Reflection f()=(-^++)/ points on the graph The point (, ) on the original graph became (, ) on the transformed graph, and the point (, ) became the point (, ). For both sets of points, the values change sign. Negative values become positive, and positive values become negative. Fortunatel, putting a sign in front of a variable changes its sign. That is, if =, then = ( ) =. And if = 0, then = 0. The sign changes negatives to positives and vice versa. To accomplish this ais reflection then, we change the signs of the values. We replace f ( ) with f ( ). Q ( ) For eample, to reflect the parabola Q ( ) over the ais, we compute, but be sure to distribute the minus sign: Q ( ) ( ) Q ( ) ais Reflection Summar To reflect a function f ( ) around the ais, compute f ( ) (Don t forget to distribute the minus sign, if necessar.) Sometimes ou ll see this as about the ais as well

7 Week Topic Transformations of Functions 7 ais reflections Since an ais reflection changes the values, it s an outside transformation we accomplish b multipling b. Based on that, ou might be able to guess that a ais reflection will be an inside transformation we accomplish b multipling b. Let s look again at the graph of the function in Figure 7. The formula is f( ). We can compute f ( ) b replacing each with ( ): f ( ) f( ) f( ) Figure shows a graph of f ( ) and f ( ) f()=(^--)/ points on the graph Figure A Function and its ais Reflection 0 f(-)=(-^+-)/ points on the graph The original graph contains the points (, ) and (, ). The transformed graph contains the points (, ) and (, ). The values had their signs changed, which resulted in a ais reflection. ais Reflection Summar To reflect a function f ( ) around the ais, compute f ( ) (Avoid minus sign errors b substituting ( ) in for.)

8 Week Topic Transformations of Functions Reflection Eample Problem. For the function R ( ), write a formula to reflect the graph of R around a. the ais. b. the ais. Reflection and Shift Problem. Let s combine reflections and shifts. Start with the absolute value function f ( ), and perform each of these transformations. Sketch a graph of each resulting function, and be sure it matches what it should look like. Do the transformations in the order listed. a. Reflect around the ais, shift up b. Reflect around the ais, shift up c. Shift down, reflect around the ais d. Shift left, reflect around the ais Check our answers with a graphing utilit. Be sure to find a point or two on the original graph to compare with each of our transformations.

9 Week Topic Transformations of Functions 9 Stretches and Compressions Vertical Stretches and Compressions When we shift a graph verticall, we add or subtract an amount to each value. A vertical stretch also affects the values, but instead of adding or subtracting, we multipl or divide the values. Figure 9 shows a cubic polnomial and a vertical stretch of it. f()=^-^- points on the graph Figure 9 A Cubic Polnomial without and with a Vertical Stretch points on the graph f()=^-^ The original graph contains the points (, ) and (, ). If we multipl the values b, then the points on the transformed graph are (, ) and (, ), as seen on the transformed graph. Ever value is multiplied b, so (, ) goes to (, ). Notice the intercept (, 0) is still at (, 0) after the transformation. That s because 0 = 0. The amount we multipl b is called the stretch factor. Factor alwas indicates multiplication, as in The prime factors of are and. The numbers and are multiplied together to get. In the eample in Figure 9, we might sa The function is stretched verticall b a factor of. If we multipl the values b ½, we d sa the graph is compressed b a factor of ½. In that case, compression is the appropriate word because the values get smaller (closer to zero). As another eample, if ou have the function n ( ), we could compute n () and n ( ) as follows:

10 Week Topic Transformations of Functions 0 () n () n n ( ) n ( ) ( ) In the first case, we sa n was stretched b a factor of. In the latter case, we sa n was compressed b a factor of one third. Figure 0 shows graphs of n ( ) and the two transformations. n()=/(+^) Figure 0 A Function with a Stretch and a Compression n()=/(+^) /*n()=/(*(+^)) You can see how multipling b makes ever value three times as big. Multipling b a third makes ever value a third of its original size. The vocabular and mechanics of this operation are ver logical. Yet man students perfectl competent with shifts and reflections completel shut down when faced with a stretch or compression. I m not sure wh, but keep in mind we have seen this before. In the formula for a quadratic equation, we have the constant a which is a shape factor. Large values of a make the parabola narrower (vertical stretch) whereas values of a less than make a wider parabola (vertical compression). And then if a is negative, the parabola opens down ( ais reflection). The same idea can be applied to linear functions. The intercept is a vertical shift, and the slope represents a vertical stretch or compression, with a slope of and intercept at (0, 0) being the identit function. Vertical Stretch and Compression Summar A function f ( ) multiplied b a positive constant a results in a vertical stretch or compression with formula af( ). If a >, it s a vertical stretch b a factor of a. If 0<a<, it s a vertical compression b a factor of a. (If a is negative, ou also have an ais reflection.)

11 Week Topic Transformations of Functions Horizontal Stretches and Compressions B this time, ou ma suspect we ll perform the same mechanics but on the inside where it applies to, and such a suspicion is correct. However, the inside transformation is once again a little more complicated. Let s compare the two functions f ( ) and g ( ). Notice g ( ) f( ).The function g will multipl each value b before taking the square root. So while f (), g().. The same input gives a larger output for g. With a little math, we can find that g. So to get a as output, we put in a smaller value. Figure 0 shows the graphs of these two functions. Figure 0 The Square Root Function and a Horizontal Compression f()=sqrt() points on f g()=sqrt() points on g (9,) (,) (9/,) (/,) Notice what happens to the two marked points. The square root of 9 is. To get that same value of out of the transformed function, we have to put in a smaller value, 9/. Similarl, if we computed h ( ), then the points (, ) and (0, ) are on the graph of h. The two points shown in the first graph of Figure 0 move further out on the ais. This is a horizontal stretch. Let s look at another eample. Take the quadratic function f( ). This function has horizontal intercepts at (, 0) and (, 0). Those intercepts will be multiplied (or divided) b a constant if we horizontall stretch or compress the graph. Figure shows the graph of f.

12 Week Topic Transformations of Functions Figure The Graph of a Quadratic with Zeroes at = and f()=(-)(+) If we compute f( ) - - -, we ll get a horizontal compression b a factor of. Indeed, b the Zero Product Principle, we can see this new function has zeroes where ( ) = 0 and ( + ) = 0, or at =, =. That is definitel a compression. If we compute f, we ll get a horizontal stretch b a factor of. Similarl to above, the new zeroes are at =, =. Figure compares these transformations. f()=(-)(+) Figure A Horizontal Compression and a Stretch of a Parabola f(/)=(/-)(/+) Horizontal Stretch and Compression Summar A function f ( ) transformed to f ( a ) (a is a positive constant) results in a horizontal stretch or compression If a >, it s a horizontal compression b a factor of a. If 0<a<, it s a horizontal stretch b a factor of a. (If a is negative, ou also have a ais reflection.)

13 Week Topic Transformations of Functions Stretch and Compression Eample Problems. Let s use the cubic polnomial function f ( ), with the graph shown below. f()=^ a. Sketch a graph of this function if it s verticall stretched b a factor of. Then, write a formula for that function c. Sketch a graph of this function if it s horizontall stretched b a factor of. Then, write a formula for that function b. First, sketch a graph of this function if it s verticall compressed b a factor of. Then, write a formula for that function b. Sketch a graph of this function if it s horizontall compressed b a factor of. Then, write a formula for that function

14 Week Topic Transformations of Functions Transformations Together Each tpe of transformations affects solel the values or solel the values. So if ou have two transformations, one inside () and one outside (), then order won t matter. However, if ou have two inside transformations or two outside transformations, order does matter. Let s illustrate that with the square root function f ( ). I ll appl two inside transformations: A horizontal shift left units, and a ais reflection, in both orders. Figure The Horizontal Shift, and then the ais Reflection Original Function Shifted left units Add in the ais reflection f ( ) f( ) f( ) Figure The ais Reflection, and then the Horizontal Shift Original Function f ( ) Reflect over the ais f ( ) Shift that graph units left f( )

15 Week Topic Transformations of Functions In the second step of Figure, we replaced with ( + ), and the sign staed out front of the parentheses. We could distribute the sign now, if we wanted to. But either wa, we end up with a different formula than the series of transformations in Figure led to. Conclusions There s plent more I could sa about transformations, but reall it comes down to practice and plent of reminding ourself how each individual transformation works. And, as alwas, work carefull to avoid algebra errors. In general, if ou re tring to get a particular curve to look a certain wa, it s best to do the compressions and stretches first, and then the shifts last. If ou do the shift first, then stretching it afterwards will alter how much it s shifted b. On a final note, notice that transformations aren t necessaril unique. What that means is more than one series of transformations might result in the same graph. For eample, if we have the square function f ( ), then a vertical stretch b a factor of 9 produces: 9 f ( ) 9 Whereas a horizontal compression b a factor of produces: f ( ) f ( ) 9 Same result, but two different was to arrive at it. Best wishes in our own personal transformations (whether the are related to functions or not).