Shifting and scaling: Starting with the grap?h ?of f ? ?? ) = ?x2, sketch the graph of the following functions. Use a graphing calculator only to check your work. ? ? a. ?f(x ? + 3) ? ? b. 2f? (? ? 4) ? ? c. ?f? (3?x) ? ? d. f? (2(? ? 3))

Step-by-step solution 14RE Step 1 To solve the problem, we first graph the function f(x) = x which is the basis for all the graphs of f(x + 3), 2f(x 4), f(3x), and f(2(x 3)). With this, the 2 graph of f(x) = x is shown below. Step 2 a) We need to graph the function f(x + 3) using the graph f(x) = x . 2 Since the given function is of the form f(x + c) with c > 0, we shift the graph of f(x) a distance c units to the left. With this, the graph of f(x + 3) is shifted horizontally left by 3 units with respect 2 to the graph of f(x) = x as shown below. Step 3 b) We need to graph the function 2f(x 4) using the graph f(x) = x . 2 Since the given function is of the form af(x c) with a, c > 0, we stretch the graph of f(x) vertically by a factor of a and shift it a distance c units to the right. With this, the graph of 2f(x 4) is scaled vertically by a factor of 2 (steepened), and is shifted horizontally right by 4 units with respect to the graph of f(x) = x 2 as shown below.