FRIDAY: Announcements TODAY ends the 2 nd week of this 5 week grading period Passing back Quiz #2 today Tuesday is Quiz #3 Thursday is your first UNIT TEST (60%)

Calendar Key Links Parent Function Packet answers Factoring Packets (Answer Keys) All notes for this unit so far Quiz Correction Forms (Precalc Tab)

Quiz Corrections Correct any problems you missed (except bonus) Due on test day!! Show all work for the reworked problems. Don’t just give a new answer! Graded for accuracy based on –How many were wrong –How many did you fix –How many were correct

Graph: f(x) = 0

Graph: f(x) = x

Graph: f(x) = x 2

Graph: f(x) =

Factoring Cubes

Greatest Integer Function (GIF) and Transformations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Objectives I can find functional values of the Greatest Integer Function (GIF) I can graph the Greatest Integer Function I can identify characteristics of the GIF I can recognize the order of transformations

NEW function page GREATEST INTEGER FUNCTION (GIF) The greatest integer function is a piece-wise defined functionpiece-wise defined function. The GIF is like the bill for your cell phone, but in reverse. If you talk for 4 ½ minutes you get charged for 4 minutes.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Greatest Integer Function f(x) = [x] or sometimes f(x) = [[x]] This generates the greatest integer less than or equal to the value of x Examples: [2.7] = 2 [-3.6] = -4 [1/3] = 0

Remembering GIF Use a number line! ALWAYS round Down 14

Start with a dark circle on the origin. The dark horizontal line is 1 unit long. It has an open circle on the right.

Greatest Integer Function The domain of this function is all real numbers. The range is all integers (Z) Would the absolute value function be even or odd or neither?

Transformations Review from Algebra-2 Types - Translations (left, right, up, down) - Reflections (x-axis, y-axis) - Size Changes (dilations) 17

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Transformation Rules EquationHow to obtain the graph For (c > 0) y = f(x) + c Shift graph y = f(x) up c units y = f(x) - c Shift graph y = f(x) down c units y = f(x – c) Shift graph y = f(x) right c units y = f(x + c) Shift graph y = f(x) left c units

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Transformation Rules EquationHow to obtain the graph y = -f(x) (c > 0)Reflect graph y = f(x) over x-axis y = f(-x) (c > 0)Reflect graph y = f(x) over y-axis y = af(x) (a > 1)Stretch graph y = f(x) vertically by factor of a y = af(x) (0 < a < 1)Compress graph y = f(x) vertically by factor of a Multiply y-coordinates of y = f(x) by a

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Translations Shifting of a graph vertically and/or horizontally Size does not change

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 f (x) f (x) + c +c+c f (x) – c -c If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units. Vertical Shifts If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units. x y Vertical Shifts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 h(x) = |x| – 4 Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4. f (x) = |x| x y g(x) = |x| + 3 Example: Vertical Shifts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Graphing Utility: Vertical Shifts Graphing Utility: Sketch the graphs given by –55 4 –4

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 x y y = f (x)y = f (x – c) +c+c y = f (x + c) -c-c If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. Horizontal Shifts If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units. Horizontal Shifts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 f (x) = x 3 h(x) = (x + 4) 3 Example: Use the graph of f (x) = x 3 to graph g (x) = (x – 2) 3 and h(x) = (x + 4) 3. x y g(x) = (x – 2) 3 Example: Horizontal Shifts

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Graphing Utility: Horizontal Shifts Graphing Utility: Sketch the graphs given by –56 7 –1

Copyright © by Houghton Mifflin Company, Inc. All rights reserved y 4 x x y 4 Example: Graph the function using the graph of. First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. (0, 0) (4, 2) (0, – 4) (4, –2) (– 5, –4) Example: Vertical and Horizontal Shifts (–1, –2)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 y = f (–x) y = f (x) y = –f (x) The graph of a function may be a reflection of the graph of a basic function. The graph of the function y = f ( – x) is the graph of y = f (x) reflected in the y-axis. The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x- axis. x y Reflection in the y-Axis and x-Axis.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x 2 in the x-axis. Example: The graph of y = x is the graph of y = x 2 shifted upward three units. This is a vertical shift. x y y = –x 2 y = x y = x 2 Example: Shift, Reflection

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 x y 4 4 y = x 2 y = – (x + 3) 2 Example: Graph y = –(x + 3) 2 using the graph of y = x 2. First reflect the graph in the x-axis. Then shift the graph three units to the left. x y – y = – x 2 (–3, 0) Example: Reflections

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31 Vertical Stretching and Compressing If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c. If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) compressed vertically by c. Example: y = 2x 2 is the graph of y = x 2 stretched vertically by 2. – 4– 4 x y 4 4 y = x 2 is the graph of y = x 2 compressed vertically by. y = 2x 2 Vertical Stretching and Shrinking

Copyright © by Houghton Mifflin Company, Inc. All rights reserved x y 4 4 y = |x| y = |2x| Horizontal Stretching and Shrinking If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by c. If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by c. Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2. is the graph of y = |x| stretched horizontally by. Example: Vertical Stretch and Shrink

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33 Graphing Utility: Vertical Stretch and Shrink Graphing Utility: Sketch the graphs given by –5 5 5

Copyright © by Houghton Mifflin Company, Inc. All rights reserved x y Example: Graph using the graph of y = x 3. Step 4: x y Step 1: y = x 3 Step 2: y = (x + 1) 3 Step 3: Example: Multiple Transformations Graph y = x 3 and do one transformation at a time.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35

Homework WS 1-6