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**Audience Class of 26 10th and 11th grade regular education students**

2 students are resource and need extended time. All students have experience with inspiration and Microsoft Word.

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Family of Functions

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**OBJECTIVES Algebra Grade 9**

13. Translate between the characteristics defining a line (i.e., slope, intercepts, points) and both its equation and graph (A-2-H) (G-3-H) 15. Translate among tabular, graphical, and algebraic representations of functions and real-life situations (A-3-H) (P-1-H) (P-2-H) Grade 11/12 4. Translate and show the relationships among non-linear graphs, related tables of values, and algebraic symbolic representations (A-1-H)

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Objectives 8. Categorize non-linear graphs and their equations as quadratic, cubic, exponential, logarithmic, step function, rational, trigonometric, or absolute value (A-3-H) (P-5-H) Geometry Grade 9 26. Perform translations and line reflections on the coordinate plane (G-3-H) Grade 11/12 16. Represent translations, reflections, rotations, and dilations of plane figures using sketches, coordinates, vectors, and matrices (G-3-H)

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**Objectives Patterns, Relations, and Functions Grade 9**

40. Explain how the graph of a linear function changes as the coefficients or constants are changed in the function’s symbolic representation (P-4-H) Grade 11/12 25. Apply the concept of a function and function notation to represent and evaluate functions (P-1-H) (P-5-H) 29. Determine the family or families of functions that can be used to represent a given set of real-life data, with and without technology (P-5-H)

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Objectives Use Inspiration to state properties of the family of functions Use Microsoft Word to write a story to demonstrate the properties of a v shaped animal

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Use a graphing calculator to examine the following functions then use Microsoft Word to write a short description of what you observed. y = x y = |x| y = x2

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Graphing y = x “Turn your calculators on.” “Press on the Y= key.” “Press on the x key” (Note: that it is diagonal to the 2nd function key) “Press on the Graph key.” Students should see the following graph on their calculator:

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**Family of Functions TYPE FUNCTION SHAPE OF GRAPH LINEAR f(X) = x**

ABSOLUTE VALUE f(x) = |x| V QUADRATIC f(X) = x2 U

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**For each function, state the type of function and the shape of the graph**

f(x) = 3(x + 4) – 5 g(x) = -2|x + 4| + 7 h(x) = 1/5(x – 0) + 0 f(x) = 1/5(x - 10)2 – 4 f(x) = 3x2 + 9 g(x) = |x| + 7 h(x) = x + 4 f(x) = 5(x - 3)2 Linear, line Absolute value, V Quadratic, U

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Translating Linear Functions

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Use a graphing calculator to examine the following functions then use Microsoft Word to write a short description of what you observed. f(x) = x + 4 f(x) = x – 6 f(x) = -x + 8 F(x) = ½x - 1

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**Use inspiration to create a bubble map to describe what could happen to a linear function.**

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Translation Moving a graph from one location to another without changing the shape or size Graphs can be moved left, right, up, down, or in combinations of up and right etc.

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**Translating Linear Functions**

parent function f(x) = x Can be changed to f(x) = a(x + h) + k f(x) = a(x – h) + k f(x) = a(x + h) – k f(x) = a(x – h) - k

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**What happens to the graph f(x) = x**

When a number is added to x? When a number is subtracted from x? When a number is added to k? When a number is subtracted from k? Moves graph to the left Moves graph to the right Moves graph up Moves graph down

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**What happens to the graph f(x) = x**

5. When f(x) is multiplied by a number between 0 and 1 6. When f(x) is multiplied by a number greater than 1 7. When f(x) is opposite 5. The line is not as steep 6. The line is steeper 7. The lines changes directions

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**Speed Graphing Write the translation as an ordered pair**

Plot the translation Write the slope as a fraction Plot the slope from the translation point let the numerator be y and the denominator be x Draw your line

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**Speed Graph and describe what happens with the following**

f(x) = 3(x + 4) – 5 g(x) = -2(x + 4) + 7 h(x) = 1/5(x – 0) + 0 f(x) = 1/5(x -10) - 4

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Use Microsoft Word to write an equation and draw an example to represent each of the 3 family of functions. Call on several students to share work with the class

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**Translating absolute Value Functions and Parabolas**

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**Can Absolute value functions and quadratics be translated?**

Yes, they can be translated just like linear functions

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**General Form of Functions**

Type General Form Example Linear f(x) = a(x - h) + k f(x) = 2(x - 3) + 5 Absolute value f(x) = a|x – h| + k f(x) = ½|x + 6| + 9 Parabola f(x) = a(x - h)2 + k f(x) = -3(x - h)2 - 7 a, h and k are any real number

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Have students work in pairs to use write and draw 5 equations for absolute value functions and quadratics to show them translated in different positions. Have each group share with the class.

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**Saga of a V Shaped Animal**

You are an animal of your choice, real or make-believe, in the shape of an absolute value function. Your owner is an Algebra II student who moves you, stretches you, hugs you, and turns you upside down. Using all you know about yourself and Microsoft Word, describe what is happening to you while the Algebra II student is playing with you. You must include at least ten facts or properties of the Absolute Value Function, f(x) = a|x – h| + k in your story. Discuss all the changes in your shape as a, h, and k change from positive, negative, or zero and get smaller and larger. Discuss the vertex, the equation of the axis of symmetry, whether you open up or down, how to find the slope of the two lines that make your “Vshape,” and your domain and range. (Write a small number (e.g., 1, 2, etc.) next to each property in the story to make sure you have covered ten properties

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**A sample story would go like this:**

“I am a beautiful black and gold Monarch butterfly named Abby flying around the bedroom of a young girl in Algebra II named Sue. Sue lies in bed and sees me light on the corner of her window sill, so my (h, k) must be (0, 0) 1. I look like a “V” 2 with my vertex at my head and wings pointing at the ceiling at a 45 angle 3. My “a” must be positive one 4. I am trying to soak up the warm rays of the sun so I spread my wings making my “a” less than one 5. The sun seems to be coming in better in the middle of the window sill, so I carefully move three hops to my left so my “h” equals 3 6. My new equation is now y = .5|x + 3| 7. Sue decided to try to catch me, so I close my wings making my “a” greater than one 8. I begin to fly straight up five inches making my “k” positive five 9 and my new equation y = 2|x + 3| Then I turned upside down trying to escape her making my “a” negative 11. Sue finally decided to just watch me and enjoy my beauty. ”

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**Rubric for project 2 pts. -Answers in paragraph form**

In complete sentences with proper grammar and punctuation 2 pts Correct use of mathematical language symbols 3 pts/discussion -each property used correctly

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