College Algebra Extra Credit Worksheet

Transcription

1 College Algebra Extra Credit Worksheet Fall 011 Math W1003 (3) Corrin Clarkson Due: Thursday, October 0 th, Instructions Each section of this extra credit work sheet is broken into three parts. The first part gives a brief explanation of a technique or concept as well as a few examples. The second part is a series of practice exercises involving that technique or concept and the final part is a series of problems that you must solve to receive credit. When solving the Exercises, you may work with others and ask for help. However, you must solve the Problems on your own. Submit only your solutions to the Problems as the Exercises will not be graded. Each problem in this worksheet is worth 3 points for a total of 1 points. If you choose to complete this worksheet, it will count as a fifth section of the first exam. Thus the total number of points on the exam will change from 6 to 83 and your score will increase by the number of points that you earn on this worksheet. If your recalculated exam score is lower than the score you originally earned, your exam score will remain the same. Table 1: Point distribution Order matters Completing the square Piecewise functions Total 6 points 6 points 9 points 1 points 1

2 Graphing composite functions Composing a function f : R R with a linear function g(x) = mx + b transforms the graph of f in a predictable way. In order to understand this transformation, we write g as a composition of simpler functions: g = g g 1 where g 1 (x) = mx and g (x) = x + b. Table summarizes how the graph of f is transformed when f is composed with one of these simpler functions. Table : Transformations by linear functions (a) Adding a constant g(x) = x + b b < 0 b > 0 f g shift right by b units shift left by b units g f shift down by b units shift up by b units (b) Scaling by a constant g(x) = mx m = 1 0 < m < 1 1 < m reflect across expand horizontally compress horizontally f g the y-axis by a factor of 1 by a factor of m m reflect across compress vertically expand vertically g f the x-axis by a factor of 1 by a factor of m m.1 Order matters.1.1 Introduction The order in which functions are composed can have a significant effect on the result of the composition; putting on your socks and then your shoes is not the same thing as putting on your shoes and then your socks. Similarly, the order in which a sequence of transformations is applied to a graph can greatly affect the result. (See Example 1.) It is helpful to know which types of transformations can be reordered without effecting the resulting graph. First we break the transformations into two groups: those that act vertically and those that act horizontally. (See Table 3.)This corresponds to distinguishing between the linear functions g and G in a composition G f g. In this composition the functions G and g do not interact with one another, thus the horizontal and vertical

3 transformations act independently on the graph. In practice, this means that given a list of transformations one can always apply all the vertical transformations and then all of the horizontal transformations without changing the end result. (See Example.) Table 3: Horizonal and vertical transformations Horizontal shift right or left stretch horizontally compress horizontally reflect across the y-axis Vertical shift up or down stretch vertically compress vertically reflect across the x-axis Example 1. Consider the following two sequences of transformations: First sequence 1. reflect across the x-axis. stretch vertically by a factor of two 3. shift up by ten units Second sequence 1. shift up by ten units. reflect across the x-axis 3. stretch vertically by a factor of two The only difference between these sequences is the order of the transformations. None the less, they have completely different effects on the identity function id R (x) = x. The first sequence transforms the identity to the linear function G 1 (x) = x + 10 where as the second sequence transforms the identity into the linear function G (x) = (x + 10) = x 0. When the first sequence of transformations is a applied to the graph of f(x) = x 3 the resulting graph is that of G 1 f. Similarly, applying the second sequence of transformations to the graph of f gives the graph of G f. (See Figure 1.) 3

4 Figure 1: Reordering transformations (a) No transformations f x x 3 (b) First sequence f x 10 x 3 (c) Second sequence f x x 3 10 Example. The following two sequences of transformations have the same effect on a graph. First sequence 1. shift up by 1 unit. shift right by two units 3. stretch vertically by a factor of three 4. compress horizontally by a factor of four 5. reflect across the x-axis Second sequence 1. shift up by 1 unit. stretch vertically by a factor of three 3. reflect across the x-axis 4. shift right by two units 5. compress horizontally by a factor of four When either of these sequences of transformations is applied to the graph of a function f, the resulting graph is that of G f g where G(x) = 3(x+1) = 3x 3 and g(x) = 4x..1. Exercises Exercise 1. Consider the linear function g(x) = 1 (x + ). Write g as a composition of simple linear functions of the type described in Table. 4

5 Exercise. Let f : R R be any function and let G and g be the linear functions G(x) = x + 1 and g(x) = 1 x + 3. Find a sequence of transformations that transforms the graph of f to that of G f g. Exercise 3. When the following sequence of transformations is applied to the graph of a function f, the resulting graph is that of the function G f g where G and g are linear functions. Find G and g. 1. compress vertically by a factor of four. shift up by two units 3. reflect across the x-axis 4. stretch horizontally by a factor of three 5. shift right by one unit.1.3 Problems Problem 1. Let f : R R be any function and let G and g be the linear functions G(x) = 1 x 1 and g(x) = 4x +. Find a sequence of transformations that transforms the graph of f to that of G f 3 g. Problem. When the following sequence of transformations is applied to the graph of a function f, the resulting graph is that of the function G f g where G and g are linear functions. Find G and g. 1. compress horizontally by a factor of 5. shift up by four units 3. reflect across the x-axis 4. stretch vertically by a factor of three 5. shift right by two units 5

6 . Completing the square..1 Introduction Every quadratic polynomial function f(x) = ax +bx+c (where a, b and c are real numbers and a 0) can be written as a composition f = G F g where F (x) = x is the simplest quadratic function, and both G(x) = Mx + B and g(x) = mx + b are linear functions. Thus we can graph f by transforming the graph of F (x) = x. (See Figure a.) The technique of completing the square can be used to decompose f into G F g. Consider the following examples. Example 3. Let f(x) = x x. We complete the square by adding and subtracting 1 from f(x): f(x) = x x = x x = (x 1) 1 From this computation, we conclude that f(x) = (G F g)(x) where F (x) = x, and G(x) = g(x) = x 1. Thus the graph of f is simply that of F shifted up by one unit and to the right by one unit. (See Figure b.) Example 4. Let f(x) = 1 x + 4x 1. In order to complete the square we must first factor out the coefficient on x and then add and subtract the appropriate constant: f(x) = 1 x + 4x 1 = 1 (x + 8x) 1 = 1 (x + 8x + 4 4) 1 = 1 (x + 8x + 4) 8 1 = 1 (x + 4) 9 It follows that f(x) = (G F g)(x) where F (x) = x, G(x) = 1 x 9 and g(x) = x + 4. Thus the graph of f is that of F shifted to the left by 4 units, compressed vertically by a factor of, and then shifted down by 9 units. (See Figure c.) 6

7 Figure : Graphs of quadratic functions (a) Simplest quadratic 8 (b) Example 3 8 (c) Example F x x f x x 1 1 f x 1 x Exercises Exercise 4. Decompose f(x) = x +6x as a composition of the form G F g and then graph f by transforming the graph of F. Exercise 5. Decompose f(x) = 1 3 x x as a composition of the form G F g and then graph f by transforming the graph of F. Exercise 6. Decompose f(x) = 4x 4x + 6 as a composition of the form G F g and then graph f by transforming the graph of F. Exercise 7. Decompose f(x) = x + x + 4 as a composition of the form G F g and then graph f by transforming the graph of F...3 Problems Problem 3. Decompose f(x) = 9x + 3x 3 as a composition of the form 4 G F g and then graph f by transforming the graph of F. Problem 4. Decompose f(x) = x + 1x 14 as a composition of the form G F g and then graph f by transforming the graph of F. 7

8 .3 Piecewise functions.3.1 Introduction When graphing compositions of piecewise functions with linear functions, it is generally easier to think in terms of transforming the graph of the original piecewise function, rather than computing the result of the composition. No the less, it is important to be able to compute such a composition. The function that results from composing a piecewise function with another function is a piecewise function. However, this resulting piecewise function may be defined on different intervals than the original piecewise function. (See Example.) Example 5. Consider the piecewise function f defined below and the linear function g(x) = x x < 0 1 f(x) = x 1 0 x x x > The composition f g is the piecewise function is defined by the formula 4 g(x) < 0 1 f(g(x)) = g(x) 1 0 g(x) (g(x)) g(x) > = 4 x < 1 x 1 1 x 0 4x + 4x + 1 x > 0 By contrast the composition g f is the piecewise function defined by the formulas g(4) x < 0 g(f(x)) = g( 1 x 1) 0 x g(x ) x > = 9 x < 0 x 1 0 x x + 1 x > 8

9 .3. Exercises In each of the following exercises, f : R R is the absolute value function: { x x < 0 f(x) = x = x x 0 Exercise 8. Graph f Exercise 9. Graph f g where g(x) = x+3. Using the notation of piecewise functions write a formula for (f g)(x). Exercise 10. Graph G f where G(x) = x + 4. Using the notation of piecewise functions write a formula for (G f)(x)..3.3 Problems In each of the following problems, f : R R is the piecewise function defined by the formula x + x < f(x) = x x 1 x > Problem 5. Graph f. Problem 6. Graph f g where g(x) = 1 x 1. Using the notation of piecewise 3 functions write a formula for (f g)(x). Problem 7. Graph G f where G(x) = 4x + 5. Using the notation of piecewise functions write a formula for (G f)(x). 9