Basic Transformations Reflections Inverses Unit 2: Function Transformation Chapter 1 Section 1.1: Horizontal and Vertical Transformations A of a function alters the and an combination of the of the graph. : the relationship between a of an original graph and the transformed graph. Translations A moves the graph. The shape or orientation change. A translated graph is to the original graph. Horizontal Translation Moves the graph left or right. What are the equations for each graph? 1 P a g e
A horizontal translation of function = f () b units is written Each point (, ) on the graph of the base function is mapped to on the transformed function. This is shown using a : Note that the sign of h is to the sign in of the function. If h is positive, the graph of the function shifts to the. Eample: In = f ( 1), h =. Each point (, ) on the graph of = f () is mapped to. This is shown b the mapping rule If f () = 2, (2, 4) maps to If h is negative, the graph of the function shifts to the. Eample: In = f ( + 5), h =. Each point (, ) on the graph of = f () is mapped to. This is shown b the mapping rule If f () = 2, (2, 4) maps to What translation is the graph on the right? 2 P a g e
A vertical translation of function = f () b _ units is written. Each point (, ) on the graph of the base function is mapped to on the transformed function. This is shown using a Note that the sign of k is to the sign in of the function. If k is positive, the graph of the function shifts. Eample: In - 7 = f (), k = Each point (, ) on the graph of = f () is mapped to This is shown b the mapping rule If f () = 2, (1, 1) maps to If k is negative, the graph of the function shifts. Eample: In + 4 = f (), k = Each point (, ) on the graph of = f () is mapped to If f () = 2, (1, 1) maps to 3 P a g e
Vertical and horizontal translations ma be combined. The graph of k = f ( h) maps each point (, ) in the base function to in the transformed function. Mapping rule: Eample 1: Graph Translations of the Form k = f ( h) a) For f () =, graph + 6 = f ( 4) and give the equation of the transformed function. Solution For f () =, the transformed function + 6 = f ( 4) is represented b h = means a translation units to the k = means a translation units Mapping Rule: Sketch the graph of f() = What are some ke points to able to draw =? Perform the mapping rule to obtain ke points of the transformed graph. Add these points to our graph and draw in the lines. 4 P a g e
Eample 1: Graph Translations of the Form k = f ( h) b) For f () as shown, graph + 5 = f ( + 2). Solution The function = f () shown in the graph above will be transformed as follows: h = means a translation units to the k = means a translation units What are some ke points to able to draw = f()? (e.g., maimum and minimum values, endpoints) Map them to new coordinates under the transformation: Mapping Rule: 5 P a g e
Eample 2: Graph Translations of the Form = f ( h) + k For f () as shown, graph = f ( - 3) + 4. Solution: The function = f () shown in the graph above will be transformed as follows: h = means a translation units to the k = means a translation units What are some ke points to able to draw = f() (e.g., maimum and minimum values, endpoints) Map them to new coordinates under the transformation: Mapping Rule: 6 P a g e
Eample 3. Complete the table Eample 4: What vertical translation is applied to = 2 if the transformed graph passes through the point (2, 10)? What horizontal translation is applied to = 2 if the transformed graph passes through the point (0, 25)? 7 P a g e
Eit Card 8 P a g e
Eample 5: Determine the Equation of a Translated Function Note: It is a common convention to use a prime () net to each letter representing an point. Verif that the shapes are b comparing slopes and lengths of line segments. Identif ke points in the base function and where the are mapped to in the translation. What are the transformations? o o Horizontal: h = Vertical: k = What is the equation of the translated function g()? Eample 6. What is the equation of the image graph, g(), in terms of f() = 2? 9 P a g e
Your Turn 1.What is the equation of the image graph, g(), in terms of f() =? 2.What is the equation of the image graph, g(), in terms of f()? 10 P a g e
Section 1.2 Reflections and Stretches Part A: Reflections We have alread seen reflections in Math 2200 with quadratics. A reflection of a graph creates a mirror image in a line called the. Reflections, like translations, do not change the of the graph. An points where the function the line of reflection. However, unlike translations, reflections ma change the of the graph. Vertical reflection: Mapping rule: line of reflection: also known as a reflection in the 11 P a g e
Horizontal reflection: Mapping rule: line of reflection: also known as a reflection in the Eample 1: Graph Reflections of a Function = f () Given = f (), graph the indicated transformation on the same set of aes. Give the mapping notation representing the transformation. Identif an invariant points. a) = f ( ) = f ( ) represents a reflection of the function in the _-ais. Mapping rule: Invariant point(s): 12 P a g e
b) = -f () = -f() represents a reflection of the function in the _-ais. Mapping rule: Invariant point(s): c) = -f (-) = -f ( ) represents a reflection of the function in the _-ais. Mapping rule: Invariant point(s): 13 P a g e
Eample 2: What transformation is shown b the graphs below? A) Horizontal reflection = f(-) B) Horizontal reflection = -f() C) Vertical reflection = f(-) D) Vertical reflection = -f() 14 P a g e
Part B: Horizontal and Vertical Stretches A stretch, unlike a translation or a reflection, changes the of the graph. However, like translations, stretches do not change the of the graph. A stretch makes a function because the stretch multiplies or divides each -coordinate b a while leaving the -coordinate unchanged. shorter: taller: Note: a (or b) means not a A stretch makes a function narrower (compression) or wider (epansion) because the stretch multiplies or divides each -coordinate b a while leaving the -coordinate unchanged. wider: narrower: 15 P a g e
Eample 1: Given f(), graph 2= f(). Describe the transformation Mapping rule: Sketch the graph Eample 2:Given f(), graph = 2f() Describe the transformation Vertical stretch b a factor of 2. Mapping rule: State an invariant points State the domain and range of the transformed function How is the domain and/or range affected b the transformation 2= f()? Sketch the graph Eample 3:Given f(), graph = f(2). Eample 2: Given f(), graph = 2f(). Describe the transformation Mapping rule: Sketch the graph Eample 2:Given f(), graph = 2f() Describe the transformation Vertical stretch b a factor of 2. Mapping rule: State an invariant points State the domain and range of the transformed function Is the equation 16 P a g e New span (range) = Old Span (Range) X VS still true?
Eample 3: Given f(), graph = f(2). Describe the transformation Mapping rule: Sketch the graph Eample 2:Given f(), graph = 2f() Describe the transformation Vertical stretch b a factor of 2. Mapping rule: State an invariant points State the domain and range of the transformed function How is the domain and/or range affected b the transformation = f(2)? Eample 4:Given f(), graph Describe the transformation f 2 Mapping rule: Sketch the graph Eample 2:Given f(), graph = 2f() Describe the transformation Vertical stretch b a factor of 2. Mapping rule: State an invariant points State the domain and range of the transformed function Is the equation 17 P a g e New span (Domain) = Old Span (Domain) X HS still true?
Eample 5: Graph Vertical and Horizontal Stretches of a Function =f() A) Graph = 5f(3) on the same set of aes. a = represents a vertical stretch b a factor of. Will the new graph be shorter or taller than the graph of the base function? b = represents a horizontal stretch b a factor of. Will the new graph be wider or narrower than the graph of the base function? Give the mapping notation representing the transformation. B) Graph 2 = f(4) on the same set of aes. a = represents a vertical stretch b a factor of. Will the new graph be shorter or taller than the graph of the base function? b = represents a horizontal stretch b a factor of. Will the new graph be wider or narrower than the graph of the base function? Give the mapping notation representing the transformation. 18 P a g e
Eample 6: Write the Equation of a Transformed Function For each of the following the graph of the function = f () has been transformed b a series of stretches and/or reflections. Write the equation of the transformed function g(). Has the orientation changed (reflection)? In which direction? Has the shape changed (stretch)? In which direction? B how much? - Recall Equation: 19 P a g e
b) The base function f () is not shown. What must it be? Add it to the graph. Has the orientation changed (reflection)? In which direction? Has the shape changed (stretch)? In which direction? B how much? In this case we must set the range to values since the actual range is Equation: 20 P a g e
Eample 7. Determine the values of a and b from the graph. 21 P a g e
Section 1.3 Combining Transformations Sketch the graph of the function - k = af (b( - h)) We have worked with translations, reflections in the - and -ais, and stretches. For the most part, these transformations have been addressed independentl of one another. We will now etend our work to functions and graphs that have all tpes of transformations. When graphing the function k = af (b( h)) given the graph of = f(), the graph can be drawn using two methods: 1. The graph could be created using. The transformations are applied to to produce the transformed graph 2. The graph could be created from a generated b a : Eample 1. Using the graph of the function, = f (), shown below, graph the transformed function = 2 f (3( 1)) + 4. Method 1 In this method it is important to. 22 P a g e
Since stretches and reflections are the result of and translations are the result of, the are applied first. Appl the following transformations to each point to produce the transformed graph: 23 P a g e
Note the can be applied in an order, as long as it is before the. Similarl, the order in which the are applied is not important, as long as the are applied. Method 2: Mapping rule = 2 f (3( 1))+ 4 24 P a g e
Eample 2. Graph - 6 = 3f (2-4) given the graph of Y = f() below. Note: It is sometimes necessar to rewrite a function before it can be graphed. Before graphing - 6 = 3f (2-4), for eample, ou should write the function as. This will help correctl identif the value of h as, rather than. Method 1 horizontal stretch of vertical stretch of horizontal translation of vertical translation of 25 P a g e
- 6 = 3f (2( - 2)) Method 2 Mapping Rule: Eample3: Given the graph of the function = f () shown, sketch the graph of f 2 3 1 2 26 P a g e
Eample4: Given the function f () = 2, sketch the graph of the transformed graph g() = -3(3 + 3) + 2 Page 39-40 # 3, 5, 6, 7 a) d) f), 8b) 9e) f) 11 a) b) 27 P a g e
Finding the equation of a function, given its graph which is a translation and/or stretch of the graph of the function = f (). Section 1.3 In this section we will compare the graph of a base function with the graph of a transformed function,, and state the of the transformed function. The focus is on functions that have a domain and range. Eample 1 Find the equation of g() as a transformation of f(): We will find the first, then the and finall the 28 P a g e
To find the stretches we need to compare the domain and range of both functions. Horizontal Stretch What is the domain of each function? o The domain of f () is which has a span of units. o The domain of g() is which has a span of units. Recall: Therefore, g() has a horizontal stretch of. Vertical Stretch What is the range of each function? The range of f () is which has a span of units The range of g() is which has a span of units. Recall: Therefore, g() has a vertical stretch of. 29 P a g e
Is the graph reflected? o. o. Thus in our equation g() = af(b( h)) + k Appling these transformations to f() produces the equation which gives the graph: Compare the ke points of this graph to g() o notice that the graph must be and to produce g(), resulting in the function: 30 P a g e
EXAMPLE 2 Using the graph shown, determine the specific equation for the image of = f() in the form = af(b( h)) + k 31 P a g e
Eample 3 Page 40 #10 (Pick one) 32 P a g e
Inverse of a Relation Section 1.4 Page 44 Inverse Relations In this section we will: eplore the relationship between the of a and its determine whether a relation and its inverse are. produce the of an from the graph of the original relation of a function so that its inverse is also a function determine the equation for the given the equation for f. What is an inverse? An inverse of a relation whatever the original relation did. For eample: The inverse of opening a door is the inverse of wrapping a gift is From a mathematics perspective, think of inverse relations as all of the mathematical operations. 33 P a g e
Eamples of inverses Function Inverse f() = + 2 g() =5 h() = 3 f() = sin Consider a table of values for = + 4 and its inverse, The tables shows that the and values are. What is the mapping for the inverse? The of the and values represent an of a process. 34 P a g e
The of the function is the of the inverse, and vice versa. This leads to the relationship between the domains and ranges of a relation and its inverse. The of the function is the of the Inverse The of the function is the of the Inverse Use (, ) (, ) to create a table of values for the inverse of a function, =f() Is there a reflective smmetr when the graphs of f and the inverse are sketched on the same set of aes? What is the ais of smmetr? Are there an invariant points? Is the inverse a function? 35 P a g e
Sketch the graph of the inverse for the function sketched below using the line =. When sketching an inverse using =, keep in mind: Is the inverse a function? Sketch the graph of the inverse for the function sketched below using the line =. Is the inverse a function? 36 P a g e
How can we tell if the inverse will be a function before we draw it? Consider the last 3 eamples. What kind of line test could be used with the graph of = f () to determine if its inverse would be a function? Line Test If it is possible for a horizontal line to of a relation, then the of the relation is. 37 P a g e
Eample: Determine whether the inverse of each relation graphed here is a function, without actuall sketching it. (iii) Do # 3 Page 52 38 P a g e
Inverse Functions The inverse of a function = f () is denoted onl if the is a. The 1 is because f represents. Note: You have alread seen this notation with trigonometric functions. Eample: Consider f() = 2 + 3 Does this graph have an inverse function? Wh?. So how do we find the equation of the inverse function? 39 P a g e
To determine the inverse of a function, interchange the - and -coordinates. OR OR Finding equations of inverses: 1. 2. 3. 4. If the inverse is a function, then 40 P a g e
Going back to f() = 2 + 3 1. Let = f() 2. Interchange and 3. Solve for 4. If the inverse is a function, then = f -1 () Eample 1 Find f -1 () for 2 f( ) A) f() = + 2 B) 5 41 P a g e
Eample 2 There is a function that converts degrees Celsius to Fahrenheit: A) What is 25 o C in o F? B) Find the inverse function that converts Fahrenheit to Celsius? C) What is 0 o F in o C Page 52 Do # 5a, d, f, 6 a, b, c 42 P a g e
So what happens if the inverse of a function is not a function? We must restrict the of the so that the becomes a function. You will see this frequentl with and later with functions. For Quadratics The domain must be to obtain an inverse function because a fails the. Consider = 2 Does this graph have an inverse function? Wh? It fails the test. We must restrict the so that the inverse is a function. There are 3 things to keep in mind when restricting the domain: 1) The restricted function must pass the test. 2) We want to include the complete set of. (ALL of the ) 3) We want to include the of the graph. This is the for quadratic functions It will be the for Trig functions 43 P a g e
Consider f() = 2 Find the inverse relation: Restriction Based on the verte Lets take for = 2. This gives for the inverse. What if we took for = 2. This gives for the inverse. In general, for Quadratics The domain is. If the verte is (h, k) then we can take or, although this is not as common 44 P a g e
Eamples: 1. Find the inverse function of f() = 2 4 What restriction do we take? What is the verte? 2. Find the inverse function of f() = 2 + 6 + 2 What restriction do we take? What is the verte? Recall b/(2a) 45 P a g e
3. Find the inverse function of f() = 2 2 8 + 11 Homework: State the restricted domain for each of the following relations so that the inverse relation is a function, and write the equation of the inverse: 46 P a g e
Determine, algebraicall or graphicall, if two functions are inverses of each other. When presented with two functions, ou should be able to determine whether or not the are inverses of one another. This can be done on a that displas both functions b sketching the line and deciding if the functions are of one another. Algebraicall, ou could be given the equation representing each function. In this case determine the of the given functions, and then decide if it is to the other given function. 1. Which of the following represent a function and its inverse function? A) B) 47 P a g e
C) Match each of the equations from the first list with its inverse in the second list: 48 P a g e