Unit 2: Function Transformation Chapter 1 Basic Transformations Reflections Inverses
Section 1.1: Horizontal and Vertical Transformations A transformation of a function alters the equation and any combination of the location, shape, and orientation of the graph. Mapping: the relationship between a set of points of an original graph and the transformed graph.
Translations A translation moves the graph right, left up or down. The shape or orientation does not change. A translated graph is congruent (same size) to the original graph.
Horizontal Translation Moves the graph left or right. What are the equations for each graph? y y y x x x
A horizontal translation of function y = f (x) by h units is written y = f (x h). Each point (x, y) on the graph of the base function is mapped to (x + h, y) on the transformed function. This is shown using a mapping rule: x, y x h, y Note that the sign of h is opposite to the sign in the equation of the function.
If h is positive, the graph of the function shifts to the right. Example: In y = f (x 1), h = 1. Each point (x, y) on the graph of y = f (x) is mapped to (x + 1, y). This is shown by the mapping rule If f (x) = x 2, (2, 4) maps to (3, 4). 2,4 3,4 x, y x 1, y
If h is negative, the graph of the function shifts to the left. Example: In y = f (x + 5), h = 5. Each point (x, y) on the graph of y = f (x) is mapped to (x 5, y). This is shown by the mapping rule If f (x) = x 2, (2, 4) maps to ( 3, 4). 2,4 3,4 x, y x 5, y
What translation is the graph on the right?
A vertical translation of function y = f (x) by k units is written y - k = f (x). Each point (x, y) on the graph of the base function is mapped to (x, y + k) on the transformed function. This is shown using a mapping rule: x, y x, y k Note that the sign of k is opposite to the sign in the equation of the function.
If k is positive, the graph of the function shifts upward. Example: In y - 7 = f (x), k = 7. Each point (x, y) on the graph of y = f (x) is mapped to (x, y + 7). This is shown by the mapping rule If f (x) = x 2, (1, 1) maps to (1, 8). 1,1 1,8 x, y x, y 7
If k is negative, the graph of the function shifts downward. Example: In y + 4 = f (x), k = 4. Each point (x, y) on the graph of y = f (x) is mapped to (x, y - 4). If f (x) = x 2, (1, 1) maps to (1, -3). 1,1 1, 3
Vertical and horizontal translations may be combined. The graph of y k = f (x h) maps each point (x, y) in the base function to (x + h, y + k) in the transformed function. Mapping rule: x, y x h, y k
Example 1: Graph Translations of the Form y k = f (x h) a) For f (x) = x, graph y + 6 = f (x 4) and give the equation of the transformed function. Solution a) For f (x) = x, the transformed function y + 6 = f (x 4) is represented by y + 6 = x 4. h = means a horizontal translation units to the k = means a vertical translation units Mapping Rule: x, y x h, y k x, y x 4, y 6
Sketch the graph of f(x) = x What are some key points to able to draw y = x? Perform the mapping rule to x, y x 4, y 6 obtain key points of the transformed graph. y Add these points to your graph and draw in the lines. Key points: x
Example 1: Graph Translations of the Form y k = f (x h) b) For f (x) as shown, graph y + 5 = f (x + 2). Solution The function y = f (x) shown in the graph above will be transformed as follows: h = means a horizontal translation units to the k = means a vertical translation units
What are some key points to able to draw y = f(x)? (e.g., maximum and minimum values, endpoints) Map them to new coordinates under the transformation:y + 5 = f (x + 2) Mapping Rule: y Key points: x
Example 2: Graph Translations of the Form y = f (x h) + k y For f (x) as shown, graph y = f (x - 3) + 4. Solution: Re-write into transformational form x The function y = f (x) shown in the graph above will be transformed as follows: h = means a horizontal translation units to the k = means a vertical translation units
What are some key points to able to draw y = f(x) (e.g., maximum and minimum values, endpoints) Map them to new coordinates y under the transformation:y - 4 = f (x - 3) Mapping Rule: x y Key points: x
Example 3. Complete the table
Example 4: A) What vertical translation is applied to y = x 2 if the transformed graph passes through the point (2, 10)? B) What horizontal translation is applied to y = x 2 if the transformed graph passes through the point (0, 25)?
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Exit Card Given the graph of y = f (x), to create a mapping rule and a table of values for each of the transformations below and graph the transformed functions. (i) y + 2 = f (x - 6) (ii) y = f (x + 2) + 5 (iii)y = f (x - 4) - 7 y=f(x) (i) (ii) (iii)
Example 5: Determine the Equation of a Translated Function Note: It is a common convention to use a prime () next to each letter representing an image point. Verify that the shapes are congruent by comparing slopes and lengths of line segments. Identify key points in the base function and where they are mapped to in the translation.
What are the transformations? Horizontal: h = Vertical: k = What is the equation of the translated function g(x)? y k = f (x h)
Example 6. What is the equation of the image graph, g(x), in terms of f(x) = x 2?
Your Turn 1.What is the equation of the image graph, g(x), in terms of f(x) = x?
Your Turn 2.What is the equation of the image graph, g(x), in terms of f(x)?
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Section 1.2 Reflections and Stretches Part A: Reflections We have already seen reflections in Math 2200 with quadratics. A reflection of a graph creates a mirror image in a line called the line of reflection. Reflections, like translations, do not change the shape of the graph. Any points where the function crosses the line of reflection do not move (invariant points). However, unlike translations, reflections may change the orientation of the graph.
y = f (x) Vertical reflection: Mapping rule: (x, y) (x, y) line of reflection: x-axis also known as a reflection in the x-axis
Horizontal reflection: y = f ( x) Mapping rule: (x, y) ( x, y) line of reflection: y-axis also known as a reflection in the y-axis
Example 1: Graph Reflections of a Function y = f (x) Given y = f (x), graph the indicated transformation on the same set of axes. Give the mapping notation representing the transformation. Identify any invariant points.
y a) y = f ( x) y = f ( x) represents a reflection of the function in the _-axis. Mapping rule: x Invariant point(s):
y b) y = -f (x) y = -f (x) represents a reflection of the function in the _-axis. Mapping rule: x Invariant point(s):
y c) y = -f (-x) y = -f ( x) represents reflection of the function in the -axes. Mapping rule: x Invariant point(s):
Example 2: What transformation is shown by the graphs below? A) Horizontal reflection y = f(-x) B) Horizontal reflection y = -f(x) C) Vertical reflection y = f(-x) D) Vertical reflection y = -f(x) y x
Part B: Horizontal and Vertical Stretches A stretch, unlike a translation or a reflection, changes the shape of the graph. However, like translations, stretches do not change the orientation of the graph.
A vertical stretch makes a function shorter (compression) or taller (expansion) because the stretch multiplies or divides each y-coordinate by a constant factor while leaving the x-coordinate unchanged. x, y x, ay a is positive Note: Negative a (or b) means reflection not a negative stretch shorter: taller: 0 a 1 a 1
A horizontal stretch makes a function narrower (compression) or wider (expansion) because the stretch multiplies or divides each x-coordinate by a constant factor while leaving the y- coordinate unchanged. 1 x, y x, y b 0 b 1 wider: narrower: b 1 b is positive
Example 1:Given f(x), graph 2y= f(x). Describe the transformation Vertical stretch by a factor of ½. Mapping rule: 1 x, y x, y 2 Sketch the graph y x State any invariant points State the domain and range of the transformed function
How is the domain and/or range affected by the transformation 2y= f(x)? Original Range New Range 0,6 0,3 Span of 6 Span of 3 1 6 3 2 New span (range) = Old Span (Range) X VS
Example 2:Given f(x), graph y= 2f(x). Describe the transformation Vertical stretch by a factor of 2. Mapping rule: x, y x,2y Sketch the graph State any invariant points State the domain and range of the transformed function Is the equation New span (range) = Old Span (Range) X VS still true? y x
Example 3:Given f(x), graph y= f(2x). Describe the transformation Horizontal stretch by a factor of ½ Mapping rule: 1 x, y x, y 2 Sketch the graph State any invariant points State the domain and range of the transformed function y x
How is the domain and/or range affected by the transformation y= f(2x)? Original Domain New Domain 4,4 2,2 Span of 8 Span of 4 1 8 4 2 New span (Domain) = Old Span (Domain) X HS
Example 4:Given f(x), graph Describe the transformation Horizontal stretch by a factor of 2 Mapping rule: x, y 2 x, y Sketch the graph x y f 2 y State any invariant points State the domain and range of the transformed function Is the equation New span (Domain) = Old Span (Domain) X HS still true? x
Example 5: Graph Vertical and Horizontal Stretches of a Function y=f(x) A) Graph y = 5f(3x) on the same set of axes. a = represents a vertical stretch by a factor of. Will the new graph be shorter or taller than the graph of the base function? b = represents a horizontal stretch by 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.
Example 5: Graph Vertical and Horizontal Stretches of a Function y=f(x) B) Graph 2y = f(4x) on the same set of axes. a = represents a vertical stretch by a factor of. Will the new graph be shorter or taller than the graph of the base function? b = represents a horizontal stretch by 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.
Example 6: Write the Equation of a Transformed Function For each of the following the graph of the function y = f (x) has been transformed by a series of stretches and/or reflections. Write the equation of the transformed function g(x).
Has the orientation changed (reflection)? In which direction? Has the shape changed (stretch)? In which direction? By how much? Recall: New span (Domain) = Old Span (Domain) X HS New span (Domain) HS = Old Span (Domain) Equation:
b) The base function f (x) 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? By how much? New span (Range) = Old Span (Range) X VS In this case we must set the range to finite values since the actual range is infinite Range of f(x) = Range of g(x)= New span (Range) VS = Old Span (Range) Equation:
Example 7. Determine the values of a and b from the graph.
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Section 1.3 Combining Transformations
Sketch the graph of the function y - k = af (b(x - h)) We have worked with translations, reflections in the x- and y-axis, and stretches. For the most part, these transformations have been addressed independently of one another. We will now extend our work to functions and graphs that have all types of transformations.
When graphing the function y k = af (b(x h)) given the graph of y = f(x), the graph can be drawn using two methods: 1. The graph could be created using transformations. The transformations are applied to each point to produce the transformed graph 2. The graph could be created from a table of values generated by a mapping rule: 1 x, y x h, ay k b
Example 1. Using the graph of the function, y = f (x), shown below, graph the transformed function y = 2 f (3(x 1))+ 4. Method 1 In this method it is important to preserve the order of operations. Since stretches and reflections are the result of multiplication and translations are the result of addition, the stretches and reflections are applied first.
Apply the following transformations to each point to produce the transformed graph: horizontal stretch of 1/3 y vertical stretch of 2 reflection in the x-axis horizontal translation of 1 unit right vertical translation of 4 units up x
Note the stretches and reflections can be applied in any order, as long as it is before the translations. Similarly, the order in which the translations are applied is not important, as long as they are applied after the stretches and reflections.
Method 2: Mapping rule y = 2 f (3(x 1))+ 4 y x
Example 2. Graph y - 6 = 3f (2x - 4) given the graph of Y = f(x) below. y Note: It is sometimes necessary to rewrite a function before it can be graphed. Before graphing y - 6 = 3f (2x - 4), for example, you should write the function as y - 6 = 3f (2(x - 2)). This will help correctly identify the value of h as 2, rather than 4.
y - 6 = 3f (2(x - 2)) Method 1 horizontal stretch of 1/2 y vertical stretch of 3 no reflection horizontal translation of 2 units right vertical translation of 6 units up x
y - 6 = 3f (2(x - 2)) Method 2 Mapping Rule: y x
Given the graph of the function y = f (x) shown, sketch the graph of y 2f 3x 1 2
Example4: Given the function f (x) = x 2, sketch the graph of the transformed graph g(x) = -3(3x + 3) + 2 y x
Finding the equation of a function, given its graph which is a translation and/or stretch of the graph of the function y = f (x). Section 1.3
In this section we will compare the graph of a base function with the graph of a transformed function, identify all transformations, and state the equation and/or mapping rule of the transformed function. The focus is on functions that have a bounded domain and range.
Example 1 Find the equation of g(x) as a transformation of f(x): We will find the stretches first, then the reflections and finally the translations
To find the stretches we need to compare the domain and range of both functions. Horizontal Stretch What is the domain of each function? The domain of f (x) is [ 3, 4] which has a span of 7 units The domain of g(x) is [ 9, 5] which has a span of 14 units. Recall: New span (Domain) HS = Old Span (Domain) Therefore, g(x) has a horizontal stretch of.
Vertical Stretch What is the range of each function? The range of f (x) is which has a span of units The range of g(x) is which has a span of units. Recall: New span (Range) VS = Old Span (Range) Therefore, g(x) has a vertical stretch of.
Is the graph reflected? Yes, in the y-axis We have a horizontal reflection. Thus in our equation g(x) = af(b(x h)) + k a = 3 and b = -1/2
Applying these transformations to f(x) produces the equation y= 3f (-1/2x) which gives the graph: y x
Compare the key points of this graph to g(x): notice that the graph must be shifted 1 unit left and 4 units up to produce g(x), resulting in the function: 1 g( x ) 3f x 1 4 2
EXAMPLE 2 Using the graph shown, determine the specific equation for the image of y = f(x) in the form y= af(b(x h)) + k
Example 3 Page 40 #10 (Pick one)
Inverse of a Relation Section 1.4 Page 44
INVERSE RELATIONS In this section we will: explore the relationship between the graph of a relation and its inverse determine whether a relation and its inverse are functions. produce the graph of an inverse from the graph of the original relation restrict the domain of a function so that its inverse is also a function determine the equation for the inverse function f -1 given the equation for f.
What is an inverse? An inverse of a relation undoes whatever the original relation did. For example: The inverse of opening a door is closing the door; the inverse of wrapping a gift is unwrapping a gift. From a mathematics perspective, think of inverse relations as undoing all of the mathematical operations.
Examples of inverses Function Inverse f(x) = x + 2 f -1 (x) = x - 2 g(x) =5x h(x) = x 3 f(x) = sin x
Consider a table of values for y = x + 4 and its inverse, y = x 4, The tables shows that the x and y values are interchanged. What is the mapping for the inverse? (x, y) (y, x)
The reversal of the x and y values represent an undoing of a process. The input of the function is the output of the inverse, and vice versa. This leads to the relationship between the domains and ranges of a relation and its inverse. The Domain of the function is the Range of the Inverse The Range of the function is the Domain of the Inverse
Use (x, y) (y, x) to create a table of values for the inverse of a function, y=f(x) x y =f(x) x y Is there a reflective symmetry when the graphs of f and the inverse are sketched on the same set of axes? Yes. What is the axis of symmetry? The line y = x. Are there any invariant points? Is the inverse a function? No. It fails the vertical line test.
Sketch the graph of the inverse for the function sketched below using the line y = x. y When sketching an inverse using y = x, keep in mind: x the invariant points all other points are reflected perpendicularly across the line of reflection Is the inverse a function?
Sketch the graph of the inverse for the function sketched below using the line y = x. Is the inverse a function?
How can we tell if the inverse will be a function before we draw it? Consider the last 3 examples. y x What kind of line test could be used with the graph of y = f (x) to determine if its inverse would be a function? A Horizontal Line
Horizontal Line Test If it is possible for a horizontal line to intersect the graph of a relation more than once, then the inverse of the relation is NOT a function.
Example: Determine whether the inverse of each relation graphed here is a function, without actually sketching it.
(iii)
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Inverse Functions f 1 x The inverse of a function y = f (x) is denoted 1 y f x only if the inverse is a function. The 1 is not an exponent because f represents a function, not a variable. Note: f x 1 1 However, f x f( x) 1 f( x) You have already seen this notation with trigonometric functions. Example: q sin 1 (x), where sin(q) = x 1
y x Consider f(x) = 2x + 3 Does this graph have an inverse function? Yes! Why? It passes the horizontal line test. So how do we find the equation of the inverse function?
To determine the inverse of a function, interchange the x- and y-coordinates. Reflect in the line y = x OR x, y y, x OR y f ( x ) x f ( y ) Interchanging x and y Finding Graph of Inverse Finding Equation of Inverse
Finding equations of inverses: 1. Let y = f(x) 2. Interchange x and y 3. Solve for y 4. If the inverse is a function, then y = f -1 (x)
Going back to f(x) = 2x + 3 1. Let y = f(x) 2. Interchange x and y y 2x 3 x 2y 3 3. Solve for y 2y x 3 1 3 y x 2 2 4. If the inverse is a function, then y = f -1 (x) 1 1 3 f ( x ) x 2 2
Example 1 Find f -1 (x) for A) f(x) = x + 2 B) f( x) x 5 2
Example 2 There is a function that converts degrees Celsius to Fahrenheit F A) What is 25 o C in o F? 9 C 32 5 B) Find the inverse function that converts Fahrenheit to Celsius? C) What is 0 o F in o C
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So what happens if the inverse of a function is not a function? We must restrict the domain of the original function so that the inverse becomes a function. You will see this frequently with quadratic functions and later with trigonometric functions
For Quadratics The domain must be restricted to obtain an inverse function because a parabola fails the horizontal line test.
y x Consider y = x 2 Does this graph have an inverse function? NO! Why? It fails the horizontal line test. We must restrict the domain 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 horizontal line test. 2) We want to include the complete set of y-values. (ALL of the RANGE) 3) We want to include the centre part of the graph. This is the vertex for quadratic functions It will be the y-intercept for Trig functions
Consider f(x) = x 2 Find the inverse relation:
Restriction Based on the vertex x 0 Lets take for y = x 2. y 0 This gives for the inverse. y x y f 1 ( x ) x x
Restriction Based on the vertex x 0 What if we took for y = x 2. y 0 This gives for the inverse. y x y 1 ( ) f x x x
In general, for Quadratics The domain is restricted based on the vertex. If the vertex is (h, k) then we can take x or x common h h, although this is not as
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Examples: 1. Find the inverse function of f(x) = x 2-4
What restriction do we take? What is the vertex? 0, 4 x 0 x 0 f 1 ( x ) x 4 f 1 ( x ) x 4
2. Find the inverse function of f(x) = x 2 + 6x + 2
What restriction do we take? What is the vertex? b Recall x 2a x 3 x 3
3. Find the inverse function of f(x) = 2x 2 8x + 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:
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Determine, algebraically or graphically, if two functions are inverses of each other. When presented with two functions, you should be able to determine whether or not they are inverses of one another. This can be done on a graph that displays both functions by sketching the line y = x and deciding if the functions are mirror images of one another. Algebraically, you could be given the equation representing each function. In this case determine the equation of the inverse of one of the given functions, and then decide if it is equivalent to the other given function.
1. Which of the following represent a function and its inverse function? A) B) y y x x
BONUS: Determine the equations of both graphs C) y x
Match each of the equations from the first list with its inverse in the second list:
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