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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%)
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Calendar Key Links Parent Function Packet answers Factoring Packets (Answer Keys) All notes for this unit so far Quiz Correction Forms (Precalc Tab)
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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
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Graph: f(x) = 0
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Graph: f(x) = x
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Graph: f(x) = x 2
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Graph: f(x) =
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Factoring Cubes
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Greatest Integer Function (GIF) and Transformations
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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
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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.
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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
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Remembering GIF Use a number line! ALWAYS round Down 14
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Start with a dark circle on the origin. The dark horizontal line is 1 unit long. It has an open circle on the right.
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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?
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Transformations Review from Algebra-2 Types - Translations (left, right, up, down) - Reflections (x-axis, y-axis) - Size Changes (dilations) 17
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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
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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
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Translations Shifting of a graph vertically and/or horizontally Size does not change
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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
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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 -4 4 4 8 g(x) = |x| + 3 Example: Vertical Shifts
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Graphing Utility: Vertical Shifts Graphing Utility: Sketch the graphs given by –55 4 –4
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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
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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 -4 4 4 g(x) = (x – 2) 3 Example: Horizontal Shifts
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Graphing Utility: Horizontal Shifts Graphing Utility: Sketch the graphs given by –56 7 –1
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 -4 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)
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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.
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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 2 + 3 is the graph of y = x 2 shifted upward three units. This is a vertical shift. x y -4 4 4 -8 8 y = –x 2 y = x 2 + 3 y = x 2 Example: Shift, Reflection
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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 – 4 4 4 -4 y = – x 2 (–3, 0) Example: Reflections
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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
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32 - 4- 4 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
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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
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 34 - 4- 4 4 4 8 x y Example: Graph using the graph of y = x 3. Step 4: - 4- 4 4 4 8 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.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35
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Homework WS 1-6
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