General Functions and Graphs Section 7 Functions Graphs and Symmetry Functions can be represented both as algebraic expressions and as graphs. So far we have concentrated on algebraic operations related to functions. In the rest of this unit, we will focus on the graphs and associated operations on them. These two different representations of functions are complimentary, each reinforcing the understanding of the other. In this section you will learn the graphs of 12 common functions. By learning (memorizing) these and their associated algebraic forms, you will be able to easily sketch the graphs of related functions. If you haven t memorized these graphs or learned some of the shortcuts associated with graphing, you can always create a graph by the 3-step process below. Graphing in this way: Build a table of values using Plot the points from the table of values for x from throughout the values on a coordinate grid domain Connect the points to get the final graph Or, if you are in need of a graph and have your graphing calculator handy, you can use it. Page 41
Symmetry Some graphs, like the one above, are symmetric in one way or another. By symmetry, we mean that the graph looks the same when viewed in more than one way. If you can rotate the graph 180 degrees, pivoting on the origin, and end up with the same graph, the graph is said to be symmetric about the origin. In this case. Such functions are called odd functions. (A polynomial with only odd exponents is symmetry about the origin, hence the name odd.) If you can reflect the graph over the y-axis and end up with the same graph, the graph is said to be symmetric about the y-axis. In this case. Such functions are called even functions. (A polynomial with only even exponents is symmetric about the y-axis, hence the name even.) In addition to being able to deduce symmetry by inspecting the graph of a function, you can also do an algebraic determination. If the result of is, then the function is even; if the result is nor odd., then the function is odd; if the result is neither, then the function is neither even To determine the symmetry of about the origin. in the absence of a graph:, therefore it is odd and symmetric 12 common graphs The 12 graphs and associated functions that follow do not constitute all the functions in the mathematical world, but they are a rich and solid beginning. Each function is in its simplest form and can be considered a parent graph from which variations can/will be easily made. You can learn to associate the shapes with the expressions by pure memorization by plotting 3 points (zero, large positive, large negative) to get a general sense a combination of these two A constant function: A linear function (straight line): Absolute value function: This particular function is even. This particular function is odd. This particular function is even. Page 42
A quadratic function (parabola): The square root function: The semi-circle function (with radius 5): This particular function is even. This particular function is the inverse of for This particular function is even. A cubic function: The cube root function: A rational function: This particular function is odd. This particular function is odd and the inverse of This particular function is odd. An exponential function: A logarithmic function: The greatest integer function: (included for completeness) Page 43
. Check for understanding For the following problems you need to be able to give a name and algebraic expression to a shape, and to identify even and odd functions. 1) For each of the graphs below, name the graphic and write the associated expression. 1a) 1b) 1c) 1d) 1e) 1f) 1g) 1h) For each of the functions below, state whether it is even, odd, or neither. 1) 5) 8) 11) 1) 6) 9) 12) 1) 7) 10) 13) Page 44
General Functions and Graphs Section 7 Problems Functions Graphs and Symmetry 1) Graph by plotting points. 2) For each of the graphs below, state the name of the curve, the algebraic function, and whether the graph is even or odd. Do this problem repeatedly until you can do it quickly without referring to the text. (You should have the same proficiency with this material as with foreign language vocabulary. Without it, you will have a hard time reading, writing, and speaking mathematics.) a) b) c) Name: cube root Function: Symmetry: odd d) Name: Function: Symmetry: e) Name: Function: Symmetry: f) Name: Function: Symmetry: Name: Function: Symmetry: Name: Function: Symmetry: : Page 45
2-continued) g) h) i) Name: Function: Symmetry: : j) Name: Function: Symmetry: k) Name: Function: Symmetry: : l) Name: Function: Symmetry: Name: Function: Symmetry: Name: Function: Symmetry: For each of the following, determine whether the function is even, odd, or neither. 1. 5. 9 2. 6. 10. 3. 7. 11. 4. 8. 12. Page 46
General Functions and Graphs Section 8 Introduction to Transformations The four graphs below all look similar in that they have the same size and shape. Each represents a different function, however. All these functions are related in that each of the last three is a transformation of the first graph, the parent graph. A transformation is just another function. It is typically a general-purpose function that operates on some other function. In this and the remaining sections we will study four common transformations of functions and the associated effects on graphs and algebraic expressions. Understanding these transformations greatly facilitates graphing functions and the reverse operation of understanding the algebraic expression associated with a given graph. This section focuses on the qualitative aspects of transforming graphs and the remaining sections explain the associated algebraic operations. Translations A translation shifts the entire graph in the direction of one of the axes. The size, shape and orientation of the curve are unchanged. Doing two translations can relocate the graph to be anywhere on the coordinate plane. In the example below, the parent graph is the parabola and a single translation moves it either up, down, left, or right. Translation Parent graph Vertical (up or down) Horizontal (left or right) Page 47
Reflections A reflection flips the entire graph over either of the two axes. As with translations, the size and shape of the graph are unchanged. In the example below, the parent graph is an exponential and both reflections are shown. Reflection Parent graph Vertical (over x-axis) Horizontal (over y-axis) Stretches and shrinks The stretching and shrinking transformations proportionately change the shape of a graph. The descriptions are more complicated. For a vertical stretch/shrink, the points where the graph cross the x-axis stay fixed and all the other points of the curve move vertically, either all away from the x-axis or all toward the x- axis. For a horizontal stretch/shrink, the point where the graph crosses the y-axis stays fixed and all the other points of the curve move horizontally, either all away from the y-axis or all toward the y-axis. In the example below, the parent graph is a semicircle and the four different cases of stretching and shrinking are shown. The three dots are included in each picture to assist in comparing the pictures. Stretching and shrinking, also called dilation and compression Parent graph Vertical stretching Horizontal stretching Vertical shrinking Horizontal shrinking Page 48
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Absolute value The transformations associated with the absolute value are studied less frequently, but are presented here for completeness. For the vertical effect all negative values are reflected over the x-axis. For the horizontal effect, the graph on the right side of the y-axis is reflected over the y- axis replacing what was originally there. These transformations are shown below with two different parent functions to make their effects clear. Vertical Applied to rational function parent graph Absolute Value Horizontal Applied to exponential function parent graph Combining transformations All the examples above transform a graph just once, but a transformed graph can itself be transformed. It is a strongly recommended practice (and you will understand why later) to do horizontal transformations before vertical transformations. As an example, starting with the square root curve, (a) translate it 2 left, (b) reflect it over the y- axis, (c) translate it down 1, and then (d) stretch it vertically by a factor of 2. Start with the parent graph. a) translate 2 left a) reflect over y-axis a) translate down 1 a) stretch vertically by a factor of 2 Just for interest, the algebraic expression for this graph is (Teaser: can you see the four transformations in this expression?) Page 50
Determining transformations Consider the graph on the right. What are the transformations that generated it? To answer this question you need to identify the parent graph from the shape, identify individual transformations, and then put the transformations in an order that, starting with the parent graph, achieves the end result. a) from the shape, this is the absolute value function b) based on the vertex, it is translated right 2 c) based on the vertex, it is translated up 3 d) it is reflected over the x-axis This order doesn t work. Do it and see why. But this order works e) this is the absolute value function f) it is translated right 2 g) it is reflected over the x-axis h) it is translated up 3 The most difficult part in determining the transformations that lead to a particular graph is discerning any stretching/shrinking. To do so, you need to look at the actual coordinates of at least one point on the transformed graph to see how it compares to corresponding coordinates on the parent graph. Check for understanding For the following problems you need to be able to start with a parent graph and transform it start with a transformed curve and identify the parent and the transformations 1) Start with the cube function. Reflected it over the x-axis. Take an absolute value vertically. Translate the curve down 3. 1) Determine the parent graph and the transformations that result in the following curve. Page 51
General Functions and Graphs Section 8 Problems Introduction to transformations For problems 1-4, sketch the graph using what you know about transformations. ( Sketch means don t spend a lot of time drawing the graph. You might want to plot a couple of key points, but do not make a table of values or anything with a lot of detail.) 1) Start with the parabola. Move it right 2. Move it down 1. Reflect it over the x-axis. 2) Start with the straight line. Move it right 1. Reflect it over the y-axis. Stretch it vertically by a factor of 2. 3) Start with the straight line. Reflect it over the y-axis. Move it down 1. 4) Start with the absolute value. Move it right 1. Move it down 2. Take the absolute value vertically. Move it up 2. For each of the following curves, determine the parent graph and the associated transformations. There may be more than one correct answer. 5) 6) 7) Page 52
General Functions and Graphs Section 9 Vertical transformations For each and every transformation possible by the graphical transformations of the previous section, there is an associated algebraic operation. This section explains that relationship. For vertical translations, every (x,y) point on the original graph is shifted vertically to point (x,y +k) as shown at the right. The entire graph moves up or down k units, retaining its size, shape, and orientation. Algebraically, the function f is evaluated and then k is added to that value to produce a final value. In this particular example, the parent graph is the parabola and. The algebraic equation of the particular transformed graph is For vertical reflections, every (x,y) point on the original graph is reflected over the x-axis to point (x,-y) as shown at the right. The graph retains its size and shape. Algebraically, the function f is evaluated and is then multiplied by. In this particular example, the parent graph is the exponential. The algebraic equation of the particular transformed graph is Page 53
For vertical stretches and shrinking, every (x,y) point on the original graph is moved toward or away from the x-axis by a positive multiplier k to become (x,ky). If, the result is stretching. If, the result is shrinking. The graph proportionately changes its size and shape. Algebraically, the function f is evaluated and is then multiplied by the positive number k. In this particular example, the parent graph is a semicircle with radius 5, so and. The algebraic equation of the particular transformed graph is For the absolute value vertical effect, every (x,y) point on the original graph is repositioned to (x, y ) as shown at the right. The parts of the graph which are below the x-axis are reflected over the x-axis. Algebraically, the function f is evaluated and then all negative values are made positive to produce a final value. In this particular example, the parent graph is a rational function. The algebraic equation of the particular transformed graph is Page 54
Combining transformations and the graphical/algebraic association To a large extent, understanding the relationship between graphs and algebraic expressions is one of the major reasons for studying functions. This relationship becomes increasingly powerful when multiple transformations are applied to a single function as seen in the examples below where a parabola is the parent function. Note that the order of the transformations is critically important. 1) Reflect over the x-axis and shift up 2 1a) 1b) reflect over the x-axis yields 2b) shift up 2 2) Shift up 2 and reflect over the x-axis 2a) 3) Shift up 2, reflect over the x- axis, and take absolute value 3a) 3b) shift up 2 1c) shift up 2 yields 2c) reflect over the x-axis 3c) reflect over the x-axis 3d) take absolute value Summary With the understanding of functions, AND the conceptualization of independent and dependent variables as the function argument and the function value, AND the understanding of a few basic transformations, we can understand the relationship between algebraic expressions and graphs. For vertical transformations: Algebraically, is the value of a function, i.e., Transforming is transforming the value of a function Graphically, is plotted vertically Transforming has a vertical effect on the graph of the function Page 55
Check for understanding For the following problems you need to be able to: given any one of the three: verbal description of vertical transformations, graph, algebraic expression, quickly generate the other two. 2) Given: the cubic function shifted 2 up and reflected over the x-axis. What is the algebraic expression of this transformed function and what is a sketch of the function? 3) Given the function, describe the transformations and sketch a graph. 3) Given the graph at the right, describe the transformations and write the associated algebraic expression. Check some easily identifiable point to make sure your answer is correct. Page 56
General Functions and Graphs Section 9 Problems Vertical transformations Describe how the graph of Sketch the graph. can be transformed to the graph of the given equation. 1) 1) 3) 5) 2) 4) Describe how to transform the graph of f into the graph of g. 6) 7) For each of the following, find the equation of the reflection of f across the x-axis. 8) 9) 10) Using only vertical transformations, describe the parent graph and transformations that produce each of the following. 11) 13) 15) 12) 14) 16) Page 57
General Functions and Graphs Section 10 Horizontal transformations In the previous section, we learned that vertical transformations all correspond with changes to the value of a function. Setting shifted a graph up 2 units. We would thus expect that horizontal transformations all correspond with changes to the argument of a function. We would expect to shift a graph 2 units horizontally, and indeed it does, but in a surprising way. The graph is shifted 2 units to the left, not the right! It will help in understanding why the shift is counter-intuitive to have two different labels for the x- axis. The final result, the transformed function, has x in it, so we will leave that as is. The x-axis before the transformation will be called x (read as x prime ). For horizontal translations, every (x,y) point on the original graph is shifted horizontally to point (x=x -k,y) as shown at the right. The graph moves left or right k units, retaining its size, shape, and orientation. Algebraically, the independent variable in the argument to the function f is transformed by adding k before the function f is evaluated. In this particular example, the parent graph is the parabola and. The algebraic equation of the particular transformed graph is There are three ways to understand why this translation is to the left instead of the right. 1) Build a table of values and plot the points. If you think the result is specific to parabolas, do this for any function and you will get the same result. Page 58
2) Pick an easily identifiable point of the graph (like a vertex) and see what value of the transformed argument is associated with the vertex. a)in the parent graph, the vertex is associated with a functionn argument of 0, i.e. x=0 b)for the transformed argument to be 0, we need c)this means the vertex of the original graph moves left to 3) Consider the function evaluation sequence in the diagram at the right. The input into the function box is x. We know that like a parabola where the horizontal axis is x and the vertical axis is y, so we can graph that. But, since, looks, so the x for the transformed graph corresponds to the x for the parent graph moved left 2. For horizontal reflections, every (x,y) point on the original graph is reflected over the y- axis to point (x=-x,y) as shown at the right. The graph retains its size and shape. Algebraically, the independent variable in the argument to the function f is multiplied by before the function f is evaluated. In this particular example, the parent graph is the exponential. The algebraic equation of the particular transformed graph is Page 59
For horizontal stretches and shrinking, the result is counterintuitive again. Every (x,y) point on the original graph is toward or away from the y-axis to become (x=x /k,y). If, the result is shrinking. If, the result is stretching. The graph proportionately changes its size and shape. Algebraically, the independent variable in the argument to the function f is multiplied by k before the function f is evaluated. In this particular example, the parent graph is a semicircle with radius 5, so and The algebraic equation of the particular transformed graph is Multiplying the argument by 2 causes the graph to shrink by a factor of 2.. For the absolute value horizontal effect, every (x,y) point on the original graph is mirrored to (x=-x,y) as shown at the right. Algebraically, the independent variable in the argument to the function f is replaced by its absolute value before the function f is evaluated. In this particular example, the parent graph is the parabola. The algebraic equation of the particular transformed graph is Page 60
Combining transformations and the graphical/algebraic association As with vertical transformations, we can apply multiple horizontal transformations to a function s argument. These transformations are particularly tricky because of the counter-intuitive nature of translations and stretching/shrinking, and again order of operations is critical. 1) Reflect over the y-axis and shift left 2 1a) 1b) reflect over the y-axis yields 2b) shift left 2 2) Shift left 2 and reflect over the y-axis 2a) 3) Shift left 2, reflect over the y- axis, and take horizontal absolute value 3a) 3b) shift left 2 1c) shift left 2 yields 2c) reflect over the y-axis 3c) reflect over the y-axis (Note that only the independent variable, not the entire argument, is made negative.) 3d) use the absolute value of x The safest way to determine what effect something like has in relation to the parent function is to set, as we did at the beginning,, solve for x, and get which can be read normally as reflect x and shift 1 right Summary For horizontal transformations: Algebraically, is the argument of a function, i.e., Transforming is transforming the argument of a function Graphically, is plotted horizontally Transforming has a horizontal effect on the graph of the function Horizontal translations are counter-intuitive in that shifts the graph 2 left. Horizontal stretching/shrinking are counter-intuitive in that causes a horizontal shrinking by a factor of 2. Page 61
Check for understanding For the following problems you need to be able to: given any one of the three: verbal description of horizontal transformations, graph, algebraic expression, quickly generate the other two. 4) Given: the exponential function shifted 2 to the left and reflected over the y-axis. What is the algebraic expression of this transformed function and what is a sketch of the function? 5) Given the function, describe the transformations and sketch a graph. (Hint. Think of as so that.) 1) Given the graph at the right, describe the transformations and write the associated algebraic expression using horizontal transformations only. Check the some easily identifiable point to make sure your answer is correct. 2) Describe the transformations in problem 3 and write the associated algebraic expression using vertical transformations only. Check some easily identifiable point to make sure your answer is correct. Page 62
General Functions and Graphs Section 10 Problems Horizontal transformations Describe how the graph of Sketch the graph. can be transformed to the graph of the given equation. 1) 3) 5) 2) 4) Describe how to transform the graph of f into the graph of g. 6) 7) For each of the following, find the equation of the reflection of f across the y-axis. 8) 9) 10) Using only horizontal transformations, describe the parent graph and transformations that produce for each of the following. 11) 13) 15) 12) 14) 16) Page 63
For problems 17-21, sketch where 17) 20) 18) 21) Page 64
19) For problems 21-25, sketch where is sketched below 23) 21) 24) Page 65
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General Functions and Graphs Section 11 Combined Transformations We approached transformations by studying vertical and horizontal transformations separately in the previous two sections. The table below is a summary of all these individual transformations. The graph in the table is not one the parent functions we learned, but is a piecewise defined function, illustrating that these transformations apply to all functions. Transformations on the value of a function have a vertical effect on the graph of the function G(x). G(x)+2 {shifts up 2} -G(x) {reflects over x-axis} 2G(x) {stretch by a factor or 2} {reflects neg. values over x-axis. Transformations on G(x+2) {shifts the argument of a left 2} function have a horizontal effect on the graph of the function G(x). G(-x) {reflects over y-axis} G(2x) {shrink by a factor of 2} reflects x>0 over y-axis Of course, these transformations can be combined to produce combined transformations with both vertical and horizontal affects. The following table illustrates our first example of combined transformations from Section 8. The transformation a) starts with the square root curve b) translates it 2 left c) reflects it over the y-axis d) translates it down 1 e) stretches it vertically by a factor of 2 The example is described in English, in algebraic expressions, in rectangular coordinate graphs, and in flow diagrams. Page 67
a) start with the parent graph. a) translate it 2 left. a) reflect it over the y-axis a) translate it down 1 a) stretch it vertically by a factor of 2 This rightmost column above makes clear why it is important to do horizontal transformations before vertical transformations. We need to transform the argument first in order to get function values to transform. This is the order in which we would evaluate a function, e.g. k(4). Page 68
Check for understanding For the following problems you need to be able to: given any one of the three: verbal description of transformations, graph, algebraic expression, quickly generate the other two. 6) Given: the exponential function shifted 2 up and reflected over the y-axis. What is the algebraic expression of this transformed function and what is a sketch of the function? 7) Given the function, describe the transformations and sketch a graph 1) Given the graph at the right, describe the transformations and write the associated algebraic expression. Check some easily identifiable point to make sure your answer is correct. Page 69
General Functions and Graphs Section 11 Problems Combined Transformations Consider the two functions below: For each function below: (1) first sketch the graph, (2) describe in words how the new function relates to the base function, and for A(x) write the algebraic expression. 1b) 1c) 2b) 2c) Page 70
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3b) 3c) 4b) 4c) 5b) 5c) Page 72
6b) 6c) 7b) 7c) 8b) 8c) Page 73
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9b) 9c) 10b) 10c) 11b) 11c) Page 75
For each function graphed, write the equation of the function. Also describe the domain and range. 12) 14) f(x)= domain= range= 13) f(x)= domain= range= 15) f(x)= domain= range= f(x)= domain= range= Page 76
16) 18) f(x)= domain= range= 17) f(x)= domain= range= 19) f(x)= domain= range= f(x)= domain= range= Page 77
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General Functions and Graphs Section 12 Related Problems from SAT II Study Guides 1) If is equal to (A) 2f(-x) (B) f(x) (C) 4f(x) (D) 4f(x) (E) none of these 2) If x and y are elements in the set of real numbers, which is NOT a function (A) f={(x,y) } (B) f={(x,y) } 3) The graph of is equivalent to the graph of A) y=x B) y=2x C) y=2x for D) y=2x for E) y=2x for 4) If write A) the expression g[f(x)] in terms of x (C) f={(x,y) } B) (D) f={(x,y) } C) (E) f={(x,y) } D) E) none of these Page 83
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General Functions and Graphs References Bellman, Bragg, et. al., Advanced Algebra, Prentice Hall, 2001 Benson, Dodge, et. al., Algebra 2 and Trigonometry, McDougal, Little, 1991 Bittinger, Ellenbogen, Intermediate Algebra (5 th Ed.), Addison-Wesley, 1998 Brown, Dolciani, Sorgengrey, Kane, Algebra and Trigonometry, Structure and Method: Book 2, McDougal Littell, 2000 Brown, Robbins, Advanced Mathematics: An Introductory Course, Houghton Mifflin, 1978 Bryant, Karush, Nower, Saltz, College Algebra and Trigonometry, Goodyear, 1974 Collins et. al., Algebra 2: Integration, Applications, Connections, Glencoe-McGraw-Hill, 1997 Demana, Waits, Foley, Kennedy, Precalculus: Graphing, Numerical, Algebraic, Addison Wesley Longman, 2001 Dolciani, Graham, Swanson, Sharron, Algebra 2 and Trigonometry, Houghton Mifflin, 1989 Foerster, Algebra and Trigonometry: Functions and Applications, Addison-Wesley, 1994 Gordon, Yunker, Crosswhite, Vannatta, Advanced Mathematical Concepts: Precalculus with Applications, Glencoe Division, 1994 Gullberg, Jan, Mathematics: From the Birth of Numbers, W. W. Norton & Co., 1997 Kime, Clark, Explorations in College Algebra, John Wiley & Sons, 2001 Larson, Hostetler, Algebra and Trigonometry, Houghton Mifflin, 2001 Larson, Hostetler, Edwards, Precalculus with Limits, Houghton Mifflin, 1997 Larson, Hostetler, Nepture, Intermediate Algebra: Graphs and Functions, D.C. Health, 1994 Spiegel, Moyer, Shaum s College Algebra, McGraw Hill, 1997 Sullivan, Sullivan, Precalculus: Graphing and Data Analysis, Prentice Hall, 1998 Zuckerman, Algebra & Trigonometry, W. W. Norton & Co., 1980 Page 89