By definition, a translation is applied to a point or set of points. Intuitively, when you translate

Transcription

1 Translations and Scale Changes T hk, S ab, TRANSLATIONS, T hk By definition, a translation is applied to a point or set of points. Intuitively, when you translate a set of points, you are just "sliding" them to a different location. The translation can be in a horizontal direction (an idea) or in a vertical direction (a y idea) or both. We use the following notation to designate translations applied to a point: T :( y, ) ( + hy, + k) hk, the h value tells us how far and what direction (left or right) to move the point and the k value tells us how far and what direction (up or down) to move the point. Eample : Translate the point (-, ) three units left and four units up. T :(,) ( +,+ 4) = ( 4,6),4 Problem epressed in proper notation. Notice that left means negative, that's why we used -. Also, up means positive, so we used 4. So, the point (-, ) translated units to the left and 4 units up is the point (-4, 6). Eample : Translate the point (0, 5) two units right and si units down. T :(0,5) (0+,5+ 6) = (, ), 6 Problem epressed in proper notation. Notice that down means negative, that's why we used -6. Also, right means positive, so we used. So, the point (0, 5) translated units to the right and 6 units down is the point (, -). Can you see how a translation is an additive idea? The idea of translating points might seem pretty simple. In fact, it is! However, when we apply translations to an equation, things get a little more confusing. C onsider the equation y =. If we were looking at its graph, we would be looking at a set of points. If you were asked to translate the graph (a set of points) three units right and one unit down, you probably wouldn't have much trouble doing that. The problem arises when we try to show those same translations in the equation. In other words, how would the original equation, y =,change to show the translations? The parabola translated uints right and unit down. Revised October 0, 009

2 To show translations in an equation, memorize the following: Replace with minus the translation. i.e. ( translation) Replace y with y minus the translation. i.e. y ( ytranslation) Eample : Let y = three units down?. What is the new equation representing a translation of two units right and T, : y = y = ( ) or y + = ( ) Original Equation Equation representing the translated graph y = y+ = ( ) Eample : Let y = e units up?. What is the new equation representing a translation of one unit left and two ( ) ( + ) T, : y = e y = e or y = e Original Equation Equation representing the translated graph ( ) y = e + y = e Eample : The original function is y =. a. What are the translations? T, b. What is the equation of the translated graph? T, : y = Eample 4: How would you represent a translation of two down? T0, The translation is 0, since adding 0 to the values will not change them. The points would move down. Revised October 0, 009 Page

3 SCALE CHANGES, S ab By definition, a (or size change) is applied to a point or set of points. Intuitively, when you apply a to a set of points, you are just "stretching" or "shrinking" them. The can act in a horizontal direction (an idea) or in a vertical direction (a y idea) or both. We use the following notation to designate s applied to a point: S :(, ) (, ) ab, y aby the a>0 value tells us the factor to stretch or shrink the point horizontally and the b>0 value tells us the factor to stretch or shrink the point vertically. Eample : Apply a of two horizontally (the values) and a of vertically (the y values) to the point (-, 9). S, :(,9), 9 = (,) Problem epressed in proper notation. Eample : Apply a of horizontally (the values) and a of vertically (the y values) to the point (6, 5). S, 0 :(6,5) 6, 5 =, Problem epressed in proper notation. Can you see how a is a multiplicative idea? The idea of applying s to points might seem pretty simple. In fact, it is! However, when we apply s to an equation, things get a little more confusing. Consider the equation y = sin. If we were looking at the graph, we would be looking at a set of points. If you were asked to halve the values and double the y values, you probably wouldn't have much trouble doing that. The problem arises when we try to show those same s in the equation. In other words, how would the original equation, y = sin,change to show the scale changes? y = sin Transformed graph The original values have been halved while the original y values have been doubled. So, the transformed graph is half as wide and twice as tall. Revised October 0, 009 Page

4 To show s in an equation, memorize the following: Replace with over the. i.e. Replace y with y over the. i.e. y y Eample : Let y = +. What is the new equation representing a horizontal of two? Notice that there is no vertically (on the y values). That means that the y scale factor will be. (Multiplying all the y values by will not change them.), : S y = + y = + Original Equation Equation representing the scaled graph Notice we used a for the y. y = + y = + The transformed graph is twice as wide as the original graph. Note: The original graph has a zero at =. The transformed graph has a zero at =, twice the original value. Another point on the original graph would be (.5, -.5). The corresponding point on the transformed graph is (, -.5). Eample : Let y =. What is the new equation representing a vertical of? + Notice that there is no horizontally (on the values). That means that the scale factor will be. (Multiplying all the values by will not change them.) y : = = or =, S y + + y + Original Equation Equation representing the scaled graph Notice we used a for the. y = + y = + Note: The original graph has a y-intercept of -. The transformed graph has a y-intercept of -, which is of the original y-intercept. Another point on the original graph would be,. The 5 corresponding point on the transformed graph would be,. 5 The transformed graph is / times smaller vertically. Revised October 0, 009 Page 4

5 + Eample : Let y =. What is the new equation representing a vertical of + 5 and a horizontal of? y 5 5 S: y = = or y =, Original 5 + Equation 5 Equation representing the scaled graph The transformed graph is.5 times bigger vertically and 0.4 times smaller horizontally. 5 5 ( ) ( ) ( ) y = 5 + y = + 5 Note: The original graph has a maimum at 5, and the transformed graph has a maimum at, =,. Another point on the original graph is. The , 8 4 corresponding point on the transformed would be 6, =,. The minimum y value on the original is 0.5 and the minimum y value on the transformed graph is 0.5 =. 4 Revised October 0, 009 Page 5

6 PUTTING IT ALL TOGETHER Let's put the translations and s together. To do that we have to be aware that there's a difference as to which transformations we apply first. It turns out that you must always apply the s first and the translations last, otherwise this whole idea of replacing with minus the translation, replacing y with y minus the translation, replacing with over the, and replacing y with y over the y won't work. Eample : Let y cos. What is the transformed equation after applying T and S? =,4, y y4 y4 T ( S,4 ): y = cos = cos = cos or = cos ( + ), Replace with over - Replace with minus Replace y with y over / Replace y with y minus 4 ( ) y 4 = cos ( + ) ( ) The transformed graph is times bigger vertically, 0.5 times smaller horizontally, translated unit to the left, and 4 units up. y = cos For the trig students: Original period: π Transformed or new period: π = π Original Amplitude: Transformed amplitude: = Original phase shift: none Transformed phase shift: left Original vertical translation: none Transformed vertical translation: 4 Original mid-line: -ais Transformed mid-line: 7+ y = = 4 Revised October 0, 009 Page 6

7 One thing that's not been mentioned: What happens if the involves a negative number? If the is negative: The result is a reflection across the y-ais. S, would represent a horizontal of in addition to reflecting the graph across the y- ais. You're replacing with over -. If you focus on just the negative part, you're replacing with the negative of. That's a reflection over the y-ais (think about it). If the y is negative: The result is a reflection across the -ais. S, would represent a vertical of in addition to reflecting the graph across the -ais. You're replacing y with y over -. If you focus on just the negative part, you're replacing y with the negative of y. That's a reflection over the -ais (think about it). A negative with the scale factor reflects the graph over the y-ais. A negative with the y scale factor reflects the graph over the -ais. Eample : Let y =. What is the transformed equation after applying S,?, : y or S y = = y = OR values are doubled first reflect over the y ais y = y = y = reflect over the y ais first double the values y = y = Revised October 0, 009

8 Connections Between Point and Equation Transformations Original Point Transformed Point Transformation Original Equation Transformed Equation. (, y) (, ) S, y = +. (, y) (, ). (, y),6 4 y y = sin y 8= sin( ) y = 4. (, y) (, ) T y = 0.5,4 y 5. (, y), y = (, y) (, ) S y = f( ), 7. (, y),4y ( ) y = 8. (, y) (, ) y = int( ) 9. (, y) (, ) T0, 0. (, y) (, ) y = 6 int 5 y = y = () 5 y = (). (, y) + 5, y y = log 5( ) + 4 ( ). (, y) (, ) S 4 y ( ), = (, y) (, ) ( ) y = sin 5 π y = sin(+ ) π 4. (, y) (, ) y = f( ) y = f (4 ) y 5. (, y) 8, y = + 6 Revised October 0, 009 Page 8

9 6. S, / T -, 4 : (, y) o 7. T -, 4 S, / : (, y) o 8. In general, is S o T = T o S? What property is this question asking about for s and translations? 9. There's a special involved in showing two figures are similar. What is this scale change called and what is the proper way to indicate it using mathematical symbols? I n the following problems, identify the parent function and the transformation(s) using the proper mathematical symbols. Write the equation for the transformed parent function. (The GRID feature is found under FORMAT.) Revised October 0, 009 Page 9

10 Replace with minus the translation. i.e. ( translation) Replace y with y minus the translation. i.e. y ( ytranslation) ( ) Thk, : y, ( + hy, + k) WHY? This is the definition of translations. It's how we apply translations to points. If ' = + h and y '= y + k, then ' would represent the transformed and y ' would represent the transformed y. If we solve these two equations for and y, we would have = ' h and y = y' k. So to apply the translations h and k to the equation y = f ( ), we must replace in the original equation with ' hand y with y' k. Are you beginning to see where the two sentences at the top of the page come from, why they are so important, and why I've asked you to memorize them? To apply the translations, make the substitutions = ' h and y = y' k in the equation y = f ( ). We now have that Thk, : y = f ( ) y' k = f( ' h). So the transformed equation representing the translations is y' k = f ( ' h). But, ' and y' just represent and y values, so we'll rewrite the transformed equation using 's and y's. i.e. y k = f ( h) So... Thk, : y = f ( ) y k = f ( h) This is how we apply translations to equations. Replace with minus the translation. i.e. ( translation) Replace y with y minus the translation. i.e. y ( ytranslation) MEMORIZE! Revised October 0, 009 Page 0

11 Replace with over the. i.e. Replace y with y over the. i.e. y WHY? y S, :( y, ) ( a, by) This is the definition of s. It's how we apply s to points. ab If ' = a and y ' = by, then ' would represent the transformed and y ' would represent the ' y' transformed y. If we solve these two equations for and y, we would have = and y =. a b So to apply the s a and b to the equation y = f ( ), we must replace the in the original ' y ' equation with and y with. Are you beginning to see where the two sentences at the top of the a b page come from, why they are so important, and why I've asked you to memorize them?? T ' y' o apply the s, make the substitutions = and y in the equation. a = y = f b ( ) y' ' We now have that Sab, : y = f ( ) = f. So the transformed equation representing the b a y' ' s is = f. But, ' and y' just represent and y values, so we'll rewrite the b a y transformed equation using 's and y's. i.e. = f. b a y So... Sab, : y = f ( ) = f This is how we apply s to equations. b a Replace with over the. i.e. Replace y with y over the. i.e. y MEMORIZE! y Revised October 0, 009 Page