October 6, 2016 Transformations (section 1.6) Day 4 page 1 Multi-step transformations Objective: Apply transformations involving multiple steps or multiple substitutions. Upcoming: We will have a test on transformations on Thurs. 10/14 (H) or Fri. 10/17 (A) In the textbook homework, we had descriptions of transformations that had multiple steps. For example: from y = x 2 : a vertical stretch by a factor of 3, then a shift right by 4 units (#47). The most effective way to draw the outcome of a sequence of transformations such as this is to draw the original graph accurately then draw another graph after each transformation step. 1. Follow these steps to draw the graph described in problem 47 (see above). a. On the first grid, draw y = x 2. Plot at least 7 points accurately before drawing the curve. b. On the next grid, apply the transformation vertical stretch by a factor of 3. c. On the final grid, apply the transformation then a shift right by 4 units. y = x 2 graph after vertical stretch final graph after shift right
October 6, 2016 Transformations (section 1.6) Day 4 page 2 2. Textbook problem 49 described this sequence of transformations: from y = x, a shift left 2 units, then a vertical stretch by a factor of 2, and finally a shift down 4 units. Make accurate step-by-step drawings of this sequence of transformations. Plot a reasonable number of points before drawing. y = x graph after shift left graph after vertical stretch final graph after shift down
October 6, 2016 Transformations (section 1.6) Day 4 page 3 3. Textbook problem 50 describes this sequence of transformations: from y = x, a shift left 2 units, then a horizontal shrink by a factor of 2, and finally a shift down 4 units. Make accurate step-by-step drawings of this sequence of transformations (you ll be able to copy some of the pictures from problem 2). Plot a reasonable number of points before drawing. Identify what you ve drawn on each grid, with the last grid showing the final result.
October 6, 2016 Transformations (section 1.6) Day 4 page 4 4. In textbook problem 43, you were asked to identify a sequence of transformations starting from y = x 2 that produces the graph of y = 2(x 3) 2 4. Your answer should have involved three steps. Make step-by-step drawings of this sequence of transformations, and write the appropriate equation underneath each graph. starting from: y = x 2 after first step: y = after second step: y = after third step: y = 2(x 3) 2 4
October 6, 2016 Transformations (section 1.6) Day 4 page 5 Using the replace x with rules in multiple places Taking a break from drawing, here s something that came up occasionally in past homework but deserves more practice. One of yesterday s handouts had a summary of all the transformation rules. For every horizontal transformation, the rule had the form replace x with [some expression]. Sometimes a function formula will have x appearing in more than one place. In those instances it s crucial to make the same replacement everywhere that x appears. Also be careful to put parentheses around the substituted expression everywhere it appears. The following problem gives you a chance to practice this skill. 5. Transform each function formula by the specified transformation. Write your answers as function formulas. You do not have to draw these graphs. a. Starting from f(x) = x + 2 cos(x), translate to the left by 3 units. b. Starting from f(x) = x 3 3x 2, translate to the right by 5 units and up by 4 units. c. Starting from f(x) = 2 x + 2 x, horizontally stretch by a factor of 3. d. Starting from f(x) = x 2 x 12, horizontally shrink by a factor of 1/2. 6. Check your answer to each part of problem 5 by making before-and-after graphs on your graphing calculator. Sketch what you see on your calculator screen. If any picture doesn t fit the transformation description, check your work from problem 5. a. b. c. d. Homework from the textbook Finally, do these four problems from the textbook: 1.6 exercises 51-54. Make accurate drawings on graph paper using the step-by-step approach taught in today s lesson.