MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry and Transformations 2.5 Variation and Applications 2.4 Symmetry and Transformations Determine whether a graph is symmetric with respect to the x axis, the y axis, and the origin. Determine whether a function is even, odd, or neither even nor odd. Given the graph of a function, graph its transformation under translations, reflections, stretchings, and shrinkings. Symmetry Algebraic Tests of Symmetry: x axis: If replacing y with y produces an equivalent equation, then the graph is symmetric with respect to the x axis. y axis: If replacing x with x produces an equivalent equation, then the graph is symmetric with respect to the y axis. Origin: If replacing x with x and y with y produces an equivalent equation, then the graph is symmetric with respect to the origin. Test x = y 2 + 2 for symmetry with respect to the x axis, the y axis, and the origin. x axis: We replace y with y: x = y 2 + 2 x = ( y) 2 + 2 x = y 2 + 2 The resulting equation is equivalent to the original so the graph is symmetric with respect to the x axis. 1
Test x = y 2 + 2 for symmetry with respect to the x axis, the y axis, and the origin. y axis: We replace x with x: x = y 2 + 2 ( x) = y 2 + 2 x = y 2 + 2 The resulting equation is not equivalent to the original so the graph is not symmetric with respect to the y axis. continued Test x = y 2 + 2 for symmetry with respect to the x axis, the y axis, and the origin. Origin: We replace x with x and y with y: x = y 2 + 2 ( x)= ( y) 2 + 2 x = y 2 + 2 The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin. Even and Odd Functions If the graph of a function f is symmetric with respect to the y axis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f( x). Determine whether the function is even, odd, or neither. 1. If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f( x) = f(x). We see that h(x) = h( x). Thus, h is even. 2
Determine whether the function is even, odd, or neither. 2. Vertical Translation Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted up b units; the graph of y = f(x) b is the graph of y = f(x) shifted down b units. Horizontal Translation Horizontal Translation For d > 0, the graph of y = f(x d) is the graph of y = f(x) shifted right d units; the graph of y = f(x + d) is the graph of y = f(x) shifted left d units. Reflections The graph of y = f(x) is the reflection of the graph of y = f(x) across the x axis. The graph of y = f( x) is the reflection of the graph of y = f(x) across the y axis. If a point (x, y) is on the graph of y = f(x), then (x, y) is on the graph of y = f(x), and ( x, y) is on the graph of y = f( x). NOTE: Set (x d) = 0 and solve for x = d which indicates movement to the right because d > 0. Set (x + d) = 0 and solve for x = d which indicates movement to the left because d > 0 or d < 0. 3
Reflection of the graph y = 3x 3 4x 2 across the x axis. Reflection of the graph y = x 3 2x 2 across the y axis. Vertical Stretching and Shrinking The graph of y = af (x) can be obtained from the graph of y = f(x) by stretching vertically for a > 1, or shrinking vertically for 0 < a < 1. Stretch y = x 3 x vertically. Each y value of y = 2(x 3 x) is two times the corresponding y value for y = x 3 x. For a < 0, the graph is also reflected across the x axis. (The y coordinates of the graph of y = af (x) can be obtained by multiplying the y coordinates of y = f(x) by a.) 4
Shrink y = x 3 x vertically. Each y value of y = (1/10)(x 3 x) is one tenth times the corresponding y value for y = x 3 x. Stretch and reflect y = x 3 x across the x axis Each y value of y = 2(x 3 x) is two times the corresponding y value for y = x 3 x and reflected about the x axis. Horizontal Stretching or Shrinking The graph of y = f(cx) can be obtained from the graph of y = f(x) by shrinking horizontally for c > 1, or stretching horizontally for 0 < c < 1. Shrink y = x 3 x horizontally. Each x value of y = (2x) 3 (2x) is 1/2 times the corresponding x value for y = x 3 x. For c < 0, the graph is also reflected across the y axis. (The x coordinates of the graph of y = f(cx) can be obtained by dividing the x coordinates of the graph of y = f(x) by c.) For example: y 1 = f(4x) and y 2 = f(x). The point (12, 5) on y 2 becomes the point (3, 5) on y 1 because 4x = 12 or x = 3. 5
Stretch y = x 3 x horizontally. Each x value of y = (0.5x) 3 (0.5x) is 2 times the corresponding x value for y = x 3 x. Stretch horizontally and reflect y = x 3 x. Each x value of y = ( 0.5x) 3 ( 0.5x) is 2 times the corresponding x value for y = x 3 x and reflected about the x axis. 214/12. First, graph the equation and determine visually whether it is symmetric with respect to the x axis, the y axis, and the origin. Then verify your assertion algebraically. x 2 + 4 = 3y Determine if symmetric with respect to x axis y axis origin 1. 2. 3. 4. 5. 6. 6
214/22. Test algebraically whether the graph is symmetric with respect to the x axis, the y axis, and the origin. Then check your work graphically, if possible, using a graphing calculator. 2y 2 = 5x 2 + 12 214/34. Determine visually whether the function is even, odd, or neither even nor odd. even odd neither 33. 34. 35. 36. 37. 38. 215/42. Test algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator. f(x) = x + 1/x 7
215/56. Describe how the graph of the function can be obtained from one of the basic graphs on p. 203. Then graph the function by hand or with a graphing calculator. g(x) = 1 / (x 2) 215/66. Describe how the graph of the function can be obtained from one of the basic graphs on p. 203. Then graph the function by hand or with a graphing calculator. f(x) = ( x) 3 215/74. Describe how the graph of the function can be obtained from one of the basic graphs on p. 203. f(x) = x 3 4 215/84. Describe how the graph of the function can be obtained from one of the basic graphs on p. 203. f(x) = 3(x + 4) 2 3 8
The point ( 12, 4) is on the graph of y = f(x). Find the corresponding point on the graph of y = g(x). 215/86. g(x) = f(x 2) 215/88. g(x) = f(4x) Given that f(x) = x 2 + 3, match the function with a transformation of from one of A. f(x 2), B. f(x) + 1, C. 2f(x), D. f(3x) 215/94. g(x) = 9x 2 + 3 215/96. g(x) = 2x 2 + 6 215/89. g(x) = f(x) 2 215/92. g(x) = f(x) Write an equation for a function that has a graph with the given characteristics. 215/98. The shape of y = x, but shifted left 6 units and down 5 units. A graph of y = f(x) follows. No formula for f is given. In Exercises 107 114, graph the given equation. 216/108. y = (1/2)f(x) 215/104. The shape of y = x, but stretched horizontally by a factor of 2 and shifted down 5 units. 9
The graph of the function f is shown in figure (a). In Exercises 119 126, match the function g with one of the graphs (a) ( h), which follow. Some graphs may be used more than once and some may not be used at all. 216/120. g(x) = f(x) + 3 217/130. A graph of the function is shown below. Exercises 129 132 show graphs of functions transformed from this one. Find a formula for the function h(x). 10