1.3.1 An example: a horizontal shift of a graph EXAMPLE 1. Let f(x) = x3 - 3x2 = x2(x - 3). Its graph is given in Figure 1.5. Note the points A(-1. -4), B(0, 0), C(1, -2), D(2. -4), E(3, 0), and P(c, f(c)). You should think of the point P as a variable point on the graph. Let g be the function whose graph is obtained from the graph of f by a horizontal shift to the right by 1 unit. Our goals are to understand how the graphing transformation is related to a transformation of the formula of f(x), and to find a symbolic expression for g(x). We accomplish this goal through the following three activities. (i) Let A', B', C', D', E', P' be the points obtained from A, B. C, D. E. P by applying the graphing transformation. We can easily compute that these points are A'(0. -4), B' = (1, 0), C' = (2, -2), D' = (3. -4), E' = (4, 0), and P' = (c + 1, f(c)). (ii) We can verify our work in (i) by plotting the graphs of f and g on the same coordinate axes (Figure 1.5). v x(x-3) B (0,0) E (3,0) + 2 C (1.-2) -2+ A (-1.-4) 4+ D (2,-4) Figure 1.5 Figure for Example I (iii) We concentrate on the point P'(c + 1, f(c)). By definition, this is a point on the graph of g, so it can be written as (x, g(x)) for some x. Equating these two points, we have c + 1 = x and f(c) = g(x). We easily obtain g(x) = /(c) = f(x - 1). Thus, the symbolic expression for g(x) is obtained from the symbolic expression of f(x) by replacing x with x - 1. For the specific function f in this example, we obtain g(x) = (x - 1)3 - 3(x - 1)2 (which we could expand as a polynomial if we so desired).I .3.2 Exercises Graphing transformations and expressions for functions: 1. Mimic the exposition in Example I to obtain a formula for g(x). relative to each of the following graphing transformations applied to the graph ofx). (a) Shift upward by 3 units (b) Wrtical stretch by a factor of 2 (0) Horizontal stretch by a factor of % {Each point on the graph of f moves farther from the y axis in the graph ofg.) (d) Reection across the horizontal axis (e) Reection across the vertical axis For each oftheee graphing transformations, you should execute the following steps: (i) Compute the coordinates of the points A', B', C", D'. E', and P'. (ii) Copy the graph of f, and use your answer to (i) to graph g on the same coordinate axes. (iii) Take your answer for the coordinates of P', and set this ordered pair equal to (x, g(x)). Equate the first coordinates, and obtain an expression for c in terms of x. Finally, use the second coordinates to obtain a formula for g(x) (in terms of x)