1.1 Horizontal & Vertical Translations
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1 Unit II Transformations of Functions. Horizontal & Vertical Translations Goal: Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations. Determining the effects of h and k in k = f( h) on the graph of = f() Sketching the graph of k = f( h) for given values of h and k, given the graph of = f() Writing the equation of a function whose graph is a vertical and/or horizontal translation of the graph of = f() (I) Determining the effects of k in - k = f() or = f() + k on the graph of = f() Eample: Given the base function = f() (a) Use a table of values for each indicated function to produce a graph on the coordinate grid. Function Table of Values Graph f() = Base Function f() =
2 Unit II Transformations of Functions Function Table of Values Graph = f() + =f() = f() -=f() General Rule about the effect of k What impact does the value of k for = f() + k or k = f() have on the transformation of the base graph = f()? The value of k represents a This affects the but not the or
3 Unit II Transformations of Functions Sketching the graph of k = f() or = f() + k given a base graph = f() Eample: Given the base graph: (a) Identif ke points for = f() and create a table of values. = f() (b) Create a new table of values for + = f() and Sketch the graph on the grid above. + = f() Image Points The point that is the result of a transformation of a point on the original graph. net to each letter representing an image point.
4 Unit II Transformations of Functions Mapping Notation Each point (, ) on the base graph of = f() in (a) above is transformed in (b) to become the point (, ) on the graph of + = f(). Using mapping notation (, ) (, ) Mapping Relating one set of points to another set of points so each point in the original set corresponds to eactl one point in the image set. (II) Determining the effects of h in = f( h) on the graph of = f() Eample: (a) Given the base function = f() Use a table of values for each indicated function to produce a graph on the coordinate grid. Function Table of Values Graph Base Function f() = f() =
5 Unit II Transformations of Functions Function Table of Values Graph = f( ) = f( + ) General Rule about the effect of h What impact does the value of h for = f( h) have on the transformation of the base graph = f()? The value of h represents a This effects the but not the or Mapping rule (, ) (, )
6 Unit II Transformations of Functions 6 Sketching the graph of = f( h) given a base graph = f() Eample: Given the base graph = f() Create a mapping rule a table of values and sketch the graph on the grid above for = f( + ) Mapping Rule: (, ) (, )
7 Unit II Transformations of Functions 7 (III) Describing the translation of each function when compared to = f(). Eample: Epress the mapping rule and describe the translation of each function compared to = f(). (a) = f( ) (b) = f() 9 (c) = f( + ) 7 (d) = f( + )
8 Unit II Transformations of Functions 8 (IV) Sketching the graph of k = f( h) or = f( h) + k given a base graph of = f(). Eample: For each function: (i) State the mapping rule (ii) Create a table of values (iii) Graph the transformed functions (a) + = f( 6) (, ) (, ) (b) = f( + ) + (, ) (, )
9 Unit II Transformations of Functions 9 (V) Writing the equation of a function based on a transformation of the base function =f() Eample: Determine the values of h and k and write the equation of the translated graph. P. # - #,#8, #,C
10 Unit II Transformations of Functions 0. Reflections and Stretches Goal: Developing an understanding of the effects of reflections on the graphs of functions and their related equations (I) Graphing Reflections in the and -ais Reflections in the -ais: Consider the point A(, ) and plot it in the coordinate grid. If the -ais represents a mirror (or reflection line), then plot and state the coordinates of the image point A. Coordinates of A Mapping a reflected point in the -ais: Mapping the point A to A is represented b A A (, ) ( )
11 Unit II Transformations of Functions Reflections in the -ais: Consider the point A(, ) and plot it in the coordinate grid. If the -ais represents a mirror (or reflection line), then plot and state the coordinates of the image point A. Coordinates of A Mapping a reflected point in the -ais: Mapping the point A to A is represented b A A (, ) ( ) Effects of Reflections on Graphs and Equations Given the graph of = f(), sketch the graph of = f() using a mapping rule or transformations. In general = f() represents a in the ais.
12 Unit II Transformations of Functions Effects of Reflections on Graphs and Equations Given the graph of = f(), sketch the graph of = f( ) using a mapping rule or transformations. In general = f( ) represents a in the ais. Summar of Reflections Creates a mirror image through a reflection line Does NOT change the of the graph DOES change the of the graph
13 Unit II Transformations of Functions Equation of a Function from a Graph & Invariant points Remember: Invariant Points A point that remains unchanged wrt position after a transformation is applied A point on a curve that lies on the line of reflection In each graph below the function = f() and a transformed graph is provided. In each case: (a) (b) (c) (d) State the tpe of transformation The mapping rule The equation of the transformed function Identif an invariant points = f() Tpe of transformation Mapping rule (, ) ( Equation: Invariant points:
14 Unit II Transformations of Functions Tpe of transformation Mapping rule (, ) ( Equation: Invariant points:. Which of the following transformations would produce a graph with the same -intercepts as = f ()? (A) = f () (B) = f ( ) (C) = f ( + ) (D) = f () +. Which ais was the first point reflected through to get the coordinates of the second point? (i) (6, 7) and ( 6, 7) (ii) (, 7) and (, 7)
15 Unit II Transformations of Functions (II) Graphing Vertical and Horizontal Stretches Graphing Vertical Stretches: Plot the point A(, ) in the coordinate grid. Plot a point A with the same -coordinate as A and a -coordinate times the coordinate in A. Plot a point A with the same -coordinate as A and a -coordinate times the coordinate in A. - - Describe how multipling the -coordinate b a factor of or b a factor of affects the position of the image point. Mapping a verticall stretched point: Mapping the point A to A or A is represented b A A or A (, ) ( )
16 Unit II Transformations of Functions 6 Effects of Verticall Stretching on Graphs and Equations Given the base graph of = f() identif the ke points = f() (a) Generate a table of values to produce the graph of = f() or = f(). = f() (b) Generate a table of values to produce the graph of = f() or = f(). = f()
17 Unit II Transformations of Functions 7 In general = af() or represents a The value of a changes the of the graph Reflection and Stretching Given the graph of = f() and = af() or (a) (b) (c) (d) the vertical stretch whether the vertical stretch can ever be negative. the mapping rule. (, ) ( ) the equation of the function. determine: =f() (e) what effects will be on the function when the: (i) a > (ii) a <
18 Unit II Transformations of Functions 8 Effects of Horizontall Stretching on Graphs and Equations Eample: Below is the base graph = f() [or = sin()] Below is the graph of = f() [or = sin ()]. Produce the table of values and state the mapping rule Mapping Rule: (, ) ( )
19 Unit II Transformations of Functions Eample: Below is the graph of = f( ) [or = sin ( )]. Produce the table of values and state the mapping rule Mapping Rule: (, ) ( ) Eample: Below is the graph of = f( ) [or = sin ( )]. Produce the table of values and state the mapping rule Mapping Rule: (, ) ( )
20 Unit II Transformations of Functions 0 General effects of a horizontal stretch on a base graph A horizontal stretch is alwas Given the function = f(b), the mapping rule is (, ) ( ) If b < 0, the graph will be as well as reflected in the ais. HS = Effects of Vertical/Horizontal Stretching on Graphs and Equations When we have both vertical and horizontal stretching on the base base graph = f() we have to consider the effect of a and b on the graph of = af(b) or Eample: Given the graph of = f() and = af(b) determine the span of each domain and range and write the equation of the transformed graph P.8 # - #0, #, #, C, C, C
21 Unit II Transformations of Functions. Combining Transformations Goal: Sketching the graph of a transformed function b appling translations, reflections and stretches Transformations: (I) Stretches and Reflections are the result of (II) Horizontal/Vertical Translations are the result of Due to the importance of the order of operations, are applied first. (I) Sketch the graph of a function k = af(b( h)) for given values of a, b, h and k given the graph of the function = f() Eample: Describing the transformations of the function = f() based on the transformed function = f(( )) +. Horizontal stretch of Vertical stretch of Reflection in the Horizontal translation Vertical translation
22 Unit II Transformations of Functions Eample: Using the graph of the function = f(), graph the transformed function = f(( )) + Determine the mapping rule for = f() based on the transformed function = f(( )) +. (, ) Create a table of values for the base graph of = f(). = f() Create a table of values for the transformed function and graph the function on the grid above. Transformed graph = f(( )) +
23 Unit II Transformations of Functions To accuratel sketch the graph of a function of the form k = af(b( h) + k. Stretches and reflections (a and b values) should occur before translation values (h and k values) Note: It is sometimes necessar to rewrite a function before it can be graphed since the horizontal translation value can be correctl identified. Eample: Epress the mapping rule for 6 = f( 8) as a transformation of = f(). mapping rule: (, ) (
24 Unit II Transformations of Functions Eample: Given the graph of = f(), sketch the graph of + = f( ) on the same grid.
25 Unit II Transformations of Functions (II) Write the equation of a function given its graph is a translation and/or stretch of the graph of the function = f() Eample: Compare the base graph f() to the graph of the transformed function g() to identif all transformations and state the equation of the transformed function. Step I Determine the horizontal stretch (HS) and the vertical stretch (VS) of = g() b comparing the domains and ranges of = f() to = g(). Domain of = f() HS of = g() Domain of = g() Range of = f() VS of = g() Range of = g()
26 Unit II Transformations of Functions 6 Step II Consider whether the points are reflected through either the or ais. Analze the orientation of image points for = g() wrt the and aes compared to the position of corresponding base points on the graph of = f(). Reflection in -ais: Reflection in -ais: Step III Develop a mapping rule on the basis of results for stretches and reflections in steps I and steps II. (, ) ( ) Step IV Test the mapping rule from step III b taking the coordinates of one base point from = f() and determining the corresponding image coordinates for = g(). Use base point ( ) appl mapping rule (, ) ( ) to determine corresponding image point ( ). Step V Plot the corresponding image point for = g() from step IV on the grid below and analze its placement to determine the appropriate horizontal and vertical translations so that the image point will be translated to the correct position. HT = VT = Step VI Appl the results for HT and VT to complete the mapping rule then write the function = af(b( h)) + k (, ) ( ) and
27 Unit II Transformations of Functions 7 Eample: Determine the specific equation for the image of = f() in the form = af(b( h)) + k.
28 Unit II Transformations of Functions 8 P. 8 # #0, #, #, C
29 Unit II Transformations of Functions 9. Inverse of a Relation Goals: Defining an inverse relation Determining the equation of an inverse Sketching the graph of an inverse relation Determining if a relation and its inverse are functions (I) What is an Inverse Relation? E. Describe the distance and direction required to travel the route indicated below. (a) From A to B (b) From B to A B km B km A km A km Distance and direction A to B Distance and direction B to A
30 Unit II Transformations of Functions 0 NOTE: An inverse relation accomplishes things: (i) (ii) the ORDER of eecution and the OPERATION With respect to Mathematical Relations, inverse relations have: (i) (ii) a change in the ORDER of algebraic eecution and the algebraic OPERATION changes (inverse operation) (II) Determining the equation of an inverse algebraicall Procedure to attain an inverse algebraicall: change f() to switch and solve for E. Determine the inverse function for f() =
31 Unit II Transformations of Functions Describe the ORDER and algebraic OPERATION on in each function below. f() = These functions are therefore Note: An inverse function can be epressed using inverse function notation f () If f() = + then epress the inverse function using inverse function notation f ().
32 Unit II Transformations of Functions (III) Sketching the graph of an inverse relation Remember attaining an algebraic inverse: and interchanges To attain an inverse graphicall and will interchange Eample: If (0, ) lies on the graph of f() then what inverse coordinates lie on the graph of f ()? Eample: For each function f() = + and (a) create a table of values Points on f() = + 0 Points on (b) graph each function on the same grid. Sketch the reflection line What is the equation of the reflection line? Points on the graph of f() are related to the points on f () b the mapping rule (, ) ( )
33 Unit II Transformations of Functions Observations from graph above: (i) What point(s) above are invariant after the reflection? (ii) Where are these invariant points located? (iii) If the intercept of a base relation is (0, b) then the inverse coordinates for inverse relation are ( ) which represents an. (iv) If points on the graph of the base relation are located in the: First quadrant then the inverse coordinates are in the (ie. If (a, b) is in the first quadrant then (, ) is in the the ) Third quadrant then the inverse coordinates are in the (ie. If ( a, b) is in the third quadrant then (, ) is in the the ) Second quadrant then the inverse coordinates are in the (ie. If ( a, b) is in the second quadrant then (, ) is in the the )
34 Unit II Transformations of Functions How to graphicall sketch an inverse relation given a graph (I) Method I Create inverse coordinates from the base graph Identif ke points from the base relation and interchange the values of and and plot the resulting inverse coordinates (II) Method II Use a blank piece of paper to create a reflection through = Trace the graph including the and aes on a piece of paper Flip the traced graph onto the original graph so and aes are lined up Rotate the blank paper 90 so the -ais is on the -ais The inverse relation will appear as an image on the underside of the blank paper. Eample: Given the graph: (a) Create a table of values using ke points. = f() (b) Create inverse coordinates (c) Graph the inverse relation.
35 Unit II Transformations of Functions (IV) Determining if a relation and its inverse are a function Remember: Graphicall distinguishing functions Graphicall distinguishing a function to determine a one to one correspondence between domain ( values) and range ( values) b the test. Eample: Which relation represents a function? (A) (B) Eample: Given the relation, sketch the inverse relation b reflecting through the line = or b appling the mapping rule (, ) (, ) Is the inverse relation a function? What kind of line test could be used on the base graph = f() to determine if the inverse would be a function?
36 Unit II Transformations of Functions 6 Eample: Without sketching, which relation would produce an inverse function? (V) Restricting the domain of a relation to attain an inverse function Graphicall attaining an inverse function Eample: How can we graphicall restrict the domain of the base graph of = + so that the inverse is a function? Remember: The predicts whether the inverse relation would be a function. How much of the domain from the given graph could be reflected to produce an inverse function? Algebraicall
37 Unit II Transformations of Functions 7 Attaining an inverse function for a quadratic Eample: Algebraicall attain the inverse function (ie. f () ) for: (a) f() = +. Procedure to attain an inverse algebraicall: change f() to switch and solve for (b) f() = ( + ) + State the verte Determine the -intercept Sketch the graph of f(). Attain the inverse function f () and sketch
38 Unit II Transformations of Functions 8 How to attain an inverse function for a quadratic in standard form Eample: Determine the inverse function for = + +. Epress = + + in verte form = ( h) + k and sketch the graph Restrict the domain of = f() so that = f () is a function. Sketch the inverse function on the same grid. State the: (i) Restricted domain and range for = f() Domain: Range: (ii) Domain and range for = f () Domain: Range:
39 Unit II Transformations of Functions 9 (VI) Determining algebraicall or graphicall if two functions are inverses of each other Graphicall Determine if there is smmetr about the line = Eample: Which graph shows a function and its inverse? (A) (B) (C) (D)
40 Unit II Transformations of Functions 0 Algebraicall Use the procedure for attaining an inverse algebraicall on one of the two functions to distinguish if the are inverses. Eample: Determine if the functions = +, and are inverses P. # #6, #0, #, # #6, #0
41 Unit II Transformations of Functions Also State the restricted domain for each of the following relations and so that the inverse relation is a function, and write the equation of the inverse. (i) = (ii) = (iii) = 8 + (iv) f() = (v) f() = + (vi) f() = +
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