Translating Points. Subtract 2 from the y-coordinates
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1 CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that changes or moves a figure is called a transformation. In this lesson, ou eplore one tpe of transformation. Investigation: Figures in Motion Steps The triangle below has vertices (, ), (, ), and (, ). If ou add to each -coordinate, ou get (, ), (, ), and (, 4). If ou subtract from each -coordinate, ou get (, ), (, ), and (, ). The grids below show the original triangle and triangles with vertices at the transformed points. Add to the -coordinates Subtract from the -coordinates Notice that adding to the -coordinates moves the triangles up units and that subtracting from the -coordinates moves the triangle down units. Now, draw our own triangle, and move it b adding or subtracting a number from the -coordinates of the vertices. Each grid in Step on page 477 shows an original triangle and the triangle that results from adding a number to or subtracting a number from the -coordinates. See if ou can figure out what number was added or subtracted. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 7
2 Previous Net Lesson 9. Translating Points (continued) Here are the answers to Step. a. was added. b. 4 was subtracted. c. was subtracted. Steps 7 Now, ou will draw and move a polgon using our calculator. The vertices of the quadrilateral shown in Step 7 are (, ), (, ), (, ), and (, ). Follow Step 8 to enter the coordinates and draw the quadrilateral on our calculator. Define lists L and L4 so that L L and L4 L. So, L contains the original -coordinates minus, and L4 contains the original -coordinates. Graph a new quadrilateral using L for the -coordinates and L4 for the -coordinates. The vertices of the new quadrilateral are (, ), (, ), (, ), and (, ). Notice that subtracting from the -coordinates of the original quadrilateral shifts it units to the left. Follow Steps 9 and at least two more times, adding a different number to or subtracting a different number from the -coordinates each time. You should find that adding a positive number to the -coordinates shifts the figure right that man units and that subtracting a positive number from the -coordinates shifts the figure left that man units. Now, let L L and let L4 L. This subtracts from the original -coordinates and adds to the original -coordinates, shifting the quadrilateral left unit and up units as shown below. Each graphing window in Step shows the original quadrilateral and a new, transformed quadrilateral. Write definitions for L and L4 in terms of L and L that would create the transformed quadrilateral. Here are the correct rules. a. L L, L4 L b. L L, L4 L c. L L, L4 L When ou transform a figure, the result is called the image of the original figure. Horizontal and vertical transformations, like the ones ou eplored in the investigation, are called translations. You can define a translation b describing the image of a general point (, ). For eample, the translation that shifts a figure left 4 units and up units can be defined as ( 4, ). Now, read and follow along with the eample in our book. 8 Discovering Algebra Condensed Lessons Ke Curriculum Press
3 CONDENSED L E S S O N 9. Translating Graphs Previous Net In this lesson ou will translate the absolute value and squaring functions translate an eponential function learn about families of functions In the previous lesson ou translated figures on the coordinate plane. In this lesson ou will learn how to translate functions. Investigation: Translations of Functions Steps If ou substitute for in the absolute value function, ou get. You can think of this as finding f( ) when f(). Enter into Y and into Y, and graph both functions. Notice that the graph of is the graph of translated right units. The verte of an absolute value graph is the point where the function changes from decreasing to increasing or from increasing to decreasing. The verte of is (, ), and the verte of is (, ). So, the verte of, like the rest of the graph, has been translated right units. The function ( 4) or 4 translates the graph of left 4 units. To get 4, ou substitute 4 for in. Write a function to create each translation of shown in Step. Use our calculator to check our work. You should get these results. a. b. c. Steps 7 Now, ou will translate along the -ais. If ou substitute for in, ou get or (solving for ). Here are the graphs of and on the same aes. Notice that the graph of is the graph of translated up units. The verte of is (, ), which is the verte of translated up units. If ou substitute ( ) or for in, ou get or. The graph of is the graph of translated down units. Write a function to create each translation of shown in Step. Use our calculator to check our work. You should get these results. a. b. c. 4 (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 9
4 Previous Net Lesson 9. Translating Graphs (continued) Step You have seen that to translate the graph of horizontall, ou subtract a number from in the function. Subtracting a positive number translates the graph right, and subtracting a negative number translates the graph left. To translate the graph verticall, ou add a number to the entire function. Adding a positive number translates the graph up, and adding a negative number translates the graph down. EXAMPLE The same ideas ou used to translate the absolute value function can be used to translate the function. Here is the graph of and ( ). Here is the graph of and. Here is the graph of and ( ). The verte of a parabola is the point where the graph changes from decreasing to increasing or from increasing to decreasing. The vertices of the parabolas above are (, ), (, ), and (, ). Notice that the -coordinate of the verte is the value subtracted from in the function and that the -coordinate is the value added to the entire function. The functions and are eamples of parent functions. B transforming a parent function, ou can create infinitel man functions in the same famil of functions. For eample, the functions ( ) and 4 are both part of the squaring famil of functions, which has as the parent function. Now, read Eample B in our book, which shows ou how to translate an eponential function. Discovering Algebra Condensed Lessons Ke Curriculum Press
5 CONDENSED L E S S O N 9. Reflecting Points and Graphs Previous Net In this lesson ou will reflect polgons over the - and -aes reflect graphs of functions over the - and -aes write equations for graphs created b combining transformations You have studied translations transformations that slide a figure horizontall or verticall. In this lesson ou will learn about transformations that flip a figure across a line. Investigation: Flipping Graphs Steps The triangle on page 49 of our book has vertices (, ), (, ), and (, ). The grid below shows the original triangle and the triangle formed b changing the sign of the -coordinate of each verte to get (, ), (, ), and (, ). Changing the signs of the -coordinates flips a figure across the -ais, creating a mirror image of the original. You can match the original figure eactl with the image b folding the grid along the -ais. The grid below shows the original triangle and the triangle formed b changing the sign of the -coordinate of each verte to get (, ), (, ), and (, ). Changing the signs of the -coordinates flips a figure across the -ais, creating a mirror image of the original. You can match the original figure eactl with the image b folding the grid along the -ais. You change the signs of both the - and -coordinates to get (, ), (, ), and (, ). This transformation flips the figure across one ais and then across the other. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons
6 Previous Net Lesson 9. Reflecting Points and Graphs (continued) Steps If ou replace the in the function with, ou get. (This is the same as finding f( ) when f().) If ou graph both functions on our calculator, ou ll see that the graph of is the graph of flipped across the -ais. If ou replace the in with, ou get or (solving for ). [This is the same as finding f() when f().] If ou graph both functions on our calculator, ou ll see that the graph of is the graph of flipped across the -ais. For each function in Step 9, replace with, and graph both the original function and the new function. In each case ou should find that the graph of the original function is flipped across the -ais to get the graph of the new function. (Note: Because the graph of is smmetric across the -ais, it looks the same when it is flipped across the -ais. So, the graphs of and are identical.) For each function in Step 9, replace with, and graph both the original function and the new function. In each case ou should find that the graph of the original function is flipped across the -ais to get the graph of the new function. A transformation that flips a figure to create a mirror image is called a reflection. As ou discovered in the investigation, a point is reflected across the -ais when ou change the sign of its -coordinate. A point is reflected across the -ais when ou change the sign of its -coordinate. Similarl, a function is reflected across the -ais when ou change the sign of, and a function is reflected across the -ais when ou change the sign of. Read the rest of the lesson in our book. Read the eample ver carefull. It eplains how to write equations for graphs created b appling more than one transformation to the graph of a parent function. Discovering Algebra Condensed Lessons Ke Curriculum Press
7 CONDENSED L E S S O N 9.4 Previous Stretching and Shrinking Graphs Net In this lesson ou will stretch and shrink a quadrilateral stretch and shrink the graph of a function write equations for graphs formed b combining transformations You have learned about transformations that slide a figure horizontall or verticall and that flip a figure across a line. In this lesson ou ll stud a transformation that changes a figure s shape. Investigation: Changing the Shape of a Graph Steps Cop the quadrilateral on page of our book onto graph paper or enter the -coordinates into list L and the -coordinates into list L. The coordinates of the vertices are (, ), (, ), (, ), and (, ). Multipl the -coordinate of each verte b to get (, ), (, ), (, ), and (, 4). On the same grid as the original figure, draw a new quadrilateral with these new points as vertices. Or follow the instructions in our book to graph it on our calculator. As ou can see, multipling the -coordinates b stretches the figure verticall. Points above the -ais move up. Points below the -ais move down. Points on the -ais remain fied. Now, multipl the -coordinates of the vertices of the original quadrilateral b, b., and b, and draw the resulting quadrilaterals. Below are the results for. and. Multipl b. Multipl b Multipling b. shrinks the figure verticall to half its size. Multipling b stretches the figure verticall and flips it over the -ais. In general, multipling the -coordinates of a figure b a number a stretches the figure if a and shrinks the figure if a reflects the figure across the -ais if a (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons
8 Previous Net Lesson 9.4 Stretching and Shrinking Graphs (continued) Steps 8 Graph the triangle shown in Step on our calculator, putting the -coordinates in list L and the -coordinates in list L. Then, predict how the definitions in parts a and b of Step 7 will transform the triangle, and use our calculator to check our answers. Here are the results. a. The figure shrinks verticall to half its size and is flipped over the -ais. b. The figure stretches verticall to twice its size and is translated down units. In Step 8, write definitions for lists L and L4 that would create each image. Here are the answers. a. L L; L4 L b. L L; L4 L Steps 9 Enter the equation f() as Y, and graph it on our calculator. Multipl the right side of the equation b that is, find f() and enter the result, ( ), as Y. Below are the graph and table for the two functions. [,,,,, ] In the table, notice that each -value for ( ) is twice the corresponding -value for. This causes the graph to stretch. The points on the graph of ( ) are twice as far from the -ais as the corresponding points on the graph of. Multipling b causes the points above the -ais to move up and the points below the -ais to move down. Repeat the process above for. f(), f(), and f(). You should find the following: Multipling b. gives -values that are half the corresponding -values for the original function, resulting in a vertical shrink. Multipling b gives -values that are times the corresponding -values for the original function, resulting in a vertical stretch. Multipling b gives -values that are times the corresponding -values for the original function, resulting in a vertical stretch and a reflection across the -ais. Look at the graphs in Step. Use what ou have learned about stretching and shrinking graphs to write an equation for R() in terms of B() and an equation for B() in terms of R(). You should get the following results: a. R() B(); B() R() b. R() B(); B() R() Read the rest of the lesson, including the eamples, ver carefull. 4 Discovering Algebra Condensed Lessons Ke Curriculum Press
9 CONDENSED L E S S O N 9. Previous Introduction to Rational Functions Net In this lesson ou will eplore transformations of the parent function predict what the graph of a rational function looks like based on its equation In Chapter, ou learned about inverse variation. The simplest inverse variation equation is. On page of our book, read about and its graph. In this lesson ou will see how the parent function can help ou understand man other functions. Investigation: I m Tring to Be Rational Steps Graph the parent function on our calculator. The functions in Step are all transformations of. Use what ou have learned in this chapter to predict how the graph of each function will compare to the graph of. The answers are given below. a. Vertical stretch b a factor of and b. Vertical stretch b a factor of reflection across the -ais and translation up units c. Translation right units d. Translation left unit and down units Now, without graphing, describe what the graph of each function in Step 4 looks like. Use the words linear, nonlinear, increasing, and decreasing. Define the domain and range, and give equations for the asmptotes. Here are sample answers. Graph a is a vertical stretch of b a factor of, followed b a translation right 4 units; it is nonlinear and decreasing; the domain is 4, and the range is ; there are asmptotes at 4 and. Graph b is a reflection of across the -ais, followed b a translation left units and down units; it is nonlinear and increasing; the domain is, and the range is ; there are asmptotes at and. Graph c is a vertical stretch of b a factor of a (if a, the graph is also reflected across the -ais), followed b a translation h units horizontall and k units verticall; it is nonlinear and decreasing if a is positive and nonlinear and increasing if a is negative; the domain is h, and the range is k; there are asmptotes at h and k. (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons
10 Previous Net Lesson 9. Introduction to Rational Functions (continued) Functions such as 4 are called rational functions because the involve ratios of two epressions. Not all rational functions are transformations of, but the graph of an rational function shares some similarities with the graph of. Steps Graph ( ) ( ) on our calculator. Compare the graph to the graphs of the functions in Step. You should find that the graph looks like the graph of. If ou trace the graph, ou should find that there is a hole at. The hole occurs because the function is not defined at (the denominator is at this value). However, for, ( )( ) So, ( )( ) is identical to for and is undefined for. Now, graph the equation in Step 9. You should find that it has vertical asmptotes at and and a horizontal asmptote at 4. In general, the following is true about the graph of a rational function: A hole occurs at h, when the factor ( h) occurs in both the numerator and the denominator. A vertical asmptote occurs at h, if the factor ( h) occurs in the denominator but not in the numerator. A horizontal asmptote occurs at k, when k is the constant of the equation. Now, read the eample in our book, which uses a rational function to model a real-world situation. Discovering Algebra Condensed Lessons Ke Curriculum Press
11 CONDENSED L E S S O N 9.7 Transformations with Matrices Previous Net In this lesson ou will relate translations to matri addition relate reflections, stretches, and shrinks to matri multiplication In our book, read the tet before the investigation, which eplains how to use a matri to represent the vertices of a geometric figure. Investigation: Matri Transformations Steps Matri [A] 4 represents 4 a triangle with vertices ( 4, ), (, 4), and (, ). Adding the matri to matri [A] adds to each -coordinate, which translates the triangle right units: [A] In parts a c of Step, find the matri sum and graph the resulting triangle. You should find that the sums correspond to the following translations of the original triangle: a. down 4 units b. right units, down 4 units c. left units, up 4 units Now, write matri equations to represent the translations in Step. Here are the answers. a. b (continued) Ke Curriculum Press Discovering Algebra Condensed Lessons 7
12 Previous Net Lesson 9.7 Transformations with Matrices (continued) Steps 7 Cop the quadrilateral on page of our book onto graph paper. Matri [B] 7 4 gives the coordinates of the vertices, along with the coordinates of a general point, (, ). Calculating [B] multiplies each -coordinate b, which reflects the quadrilateral across the -ais: 7 7 Now, perform the multiplications in Step and graph the resulting quadrilaterals. You should get the following results: a.. The -coordinates are multiplied b. This reflects the quadrilateral across the -ais b.. The -coordinates are multiplied b.. This shrinks the quadrilateral verticall b a factor of c.. The -coordinates are multiplied b., and the -coordinates are multiplied b. This shrinks the quadrilateral horizontall b a factor of. and stretches it verticall b a factor of. 4 8 Discovering Algebra Condensed Lessons Ke Curriculum Press
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