Name
October 6, 2016
Math 4 class problem set
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 = x2: 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 = x2. 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 = x2
final graph after “shift right…”
graph after “vertical stretch…”
Name
October 6, 2016
Math 4 class problem set
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 “vertical stretch…”
graph after “shift left …”
final graph after “shift down…”
Name
October 6, 2016
Math 4 class problem set
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.
Name
October 6, 2016
Math 4 class problem set
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 = x2 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 = x2
after first step: y =
after second step: y =
after third step: y = 2(x – 3)2 – 4
Name
October 6, 2016
Math 4 class problem set
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) = x3 – 3x2, translate to the right by 5 units and up by 4 units.
c. Starting from f(x) = 2x + 2–x, horizontally stretch by a factor of 3.
d. Starting from f(x) = x2 – 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.