c Sa diyya Hendrickson

Transcription

1 Transformations c Sa diyya Hendrickson Introduction Overview Vertical and Horizontal Transformations Important Facts to Remember Naming Transformations Reflections Stretches and Compressions The Rebel The Obedient y Translations (Shifts) Graphing Transformations

2 Overview Overview Consider a very basic function y = f(), suchas: y = 2, y = p, or y = (known as base/parent functions). Transformations are created by multiplying and y by constants (a, b 2 R) or adding constants (c, d 2 R) to and y, resultingin: g() =a f[b ( + c)] + d Eercise 1: Construct the transformed function g() from f() = p, that satisfies the following conditions: I f() has been multiplied by 4. (i.e.a = 4) I has been increased by 1. (i.e.c =1) Solution: g() = 4( p + 1) Vertical and Horizontal Transformations Vertical and Horizontal Transformations Transformations caused by a and d are vertical because they are directly a ecting y = f(), (whose ais, when graphing, is vertical). Transformations caused by b and c are horizontal because they are directly a ecting, (whose ais, when graphing, is horizontal). Note: Before we can correctly identify c, b must be factored out completely. We will refer to the above form as appropriate form.

3 Important Facts to Remember 1. Only the basic operations of multiplication and addition are allowed, and must occur between a variable ( or y = f()) and a constant, not between functions. Question: What is an eample of an operation that is not allowed? Answer: eponentiation (i.e. you cannot raise or y = f() to an eponent when creating transformations). 2. Transformations do not change the original family that a function belongs to. For eample, after applying transformations to a square root function, the result should also be a kind of square root function, and there should be a resemblance among the graphs. Important Facts to Remember Eercises Determine if the following is a transformed function. If so, name the base and identify a, b, c, and d. If not, eplain why. a) y = sin b) y = p 2 6 c) y = p 2 Solution: a) No has been multiplied by the function y =sin, not a constant. b) Yes Base: y = p. First, rewrite the function in appropriate form: y = p 2( 3). Therefore,a = 1,b=2,c= 3 and d =0. c) No Eponentiation has been applied to. Notice that y = p 2 =, which no longer belongs to the square-root function family, but rather the family of absolute values.

4 Reflections Recall the appropriate form: g() =a f[b ( + c)] + d I If a is negative ) vertical reflection (Graphicallly, all y-coordinates are multiplied by 1) e.g. y = 2 Base: y = ; a = 2. Note: the 2 is unrelated to the reflection. It is only the product by 1 that counts! I If b is negative ) horizontal reflection (Graphicallly, all -coordinates are multiplied by 1) e.g. y = p 4( + 1) Base: y = p ; b = 4 Stretches and Compressions Recall the appropriate form: Note: If a > 1 ) vertical stretch If a < 1 ) vertical compression If b > 1 ) horizontal compression If b < 1 ) horizontal stretch g() =a f[b ( + c)] + d multiply y-coordinates by a multiply -coordinates by 1 b I Negative parts of a and b do not a ect stretches/ compressions. I The horizontal stretches and compressions do the opposite of what you would epect. Let s eplore why!

5 The Rebel Consider y = p and the transformation: y = p y = p p p 0=0 p 1=1 p 4=2 9=3 0/2 =0 1/2 4/2 =2 9/2 = y = p p 2 p p 2(0) = p 0=0 2(1/2) p = p 1=1 p 2(2) = p 4=2 2(9/2) = 9=3 Note: In the second table, notice that can be half of its original size (i.e compressed), and still cause the function to achieve the same y-values as the first table! This is because is getting help from b =2. Q: What would happen if b = 1 2? A: would need to be twice it s original size (i.e. stretched). The Rebel Lesson Learned Graphically we can see that y = p 2 is a horizontally compressed version of y = p. More specifically, it is compressed inwards towards the y-ais, since the -values have been scaled by a factor of 1 2 = 1 b. Moral of the Story is the independent variable, and consequently can be a rebel without consequence. Therefore, constants that directly a ect require that you get in rebel-mode and DO THE OPPOSITE! e.g. Instead of multiplying by b =2,divide!

6 The Obedient y Consider y = p and the transformation: y = 2 p y = p p p 0=0 p 1=1 p 4=2 9=3 y = p p 0 = 2(0) = p 1 = 2(1) = p 4 = 2(2) = p 9 = 2(3) = 6 Note: In the second table, notice that the y-values have doubled (i.e. stretched) as we d epect, since a =2.Here,a is helping the function grow faster for the same values of in the first table. Q: What would happen if a = 1 2? A: The y-values would be half of their original size (i.e. compressed). The Obedient y Lesson Learned Graphically we can see that y =2 p is a vertically stretched version of y = p. More specifically, it is stretched upwards since the y-values have been scaled by a factor of a =2. Moral of the Story y is the dependent variable, and consequently must be obedient. Therefore, constants directly a ecting the function y = f() allow you to FOLLOW YOUR INTUITION!

7 Translations (Shifts) Recall the appropriate form: g() =a f[b ( + c)] + d If c > 0 ) Horizontal shift c units left (subtract from -coordinates) If c < 0 ) Horizontal shift c units right (add to -coordinates) If d > 0 ) vertical shift up d units (add to y-coordinates) If d < 0 ) vertical shift down d units (subtract from y-coordinates) Note: Again, we must do the opposite in the case of horizontal shifts! Eercise: Sketch the graph of y = S1 Identify the base function and create a basic table and sketch. Why? So that you know the family that your final function belongs to! Here the final graph should still have a V -shape, turned up or down. Base: y = y = 1 1 =1 0 0 =0 1 1 =1

8 S2 Put the transformed function in appropriate form. (i.e. Factor out any constant directly in front of and simplify the epression if possible). y = = 2( 3) + 1 (factor out 2) = (by properties of absolute value) = (by definition of absolute value) S3 Identify and list your transformations by asking: Is this a direct a ect on or y? Be sure to record shifts last and identify the operations on the - and y-coordinates. From the appropriate form: ) V. reflection! multiply y-coordinates by 1. 2) V. stretch by a factor of 2! multiply y-coordinates by 2. 3) H. shift 3 units right! add 3 to the -coordinates. (rebel-mode!) 4) V. shift 1 unit up! add 1 to the y-coordinates.

9 S4 Do a quick sketch of each transformation (by building on the previous one) so that you have a general idea of where the final graph should be. S5 Use the original points of y = in S1 and your list of operations in S3 to construct the eact points for your final graph.then sketch. **Confirm that this graph corresponds to your final sketch in S4. (, y) (+3, 2y+1) ( 1, 1) (( 1) + 3, 2(1) + 1) = (2, 1) (0, 0) ((0) + 3, 2(0) + 1) = (3, 1) (1, 1) ((1) + 3, 2(1) + 1) = (4, 1)