2.1 Transformers: Shifty y s A Develop Understanding Task
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1 3 2.1 Transformers: Shifty y s A Develop Understanding Task Optima is designing a robot quilt for her new grandson. She plans for the robot to have a square face. The amount of fabric that she needs for the face will depend on the area of the face, so Optima decides to model the area of the robot s face mathematically. She knows that the area A of a square with side length is modeled by the function, A() = What is the domain of the function A() in this contet? 2. Match each statement about the area to the function that models it: Matching Statement Function Equation Equation (A,B, C, or D) The length of any given side is A) A = 5 2 increased by 5 units. The length of any given side is B) A = ( + 5) 2 multiplied by 5 units. The area of a square is increased by 5 C) A = (5) 2 square units. The area of a square is multiplied by 5. D) A = Optima started thinking about the graph of y = 2 (in the domain of all real numbers) and wondering about how changes to the equation of the function like adding 5 or multiplying by 5 affect the graph. She decided to make predictions about the effects and then check them out. 3. Predict how the graphs of each of the following equations will be the same or different from the graph of y = 2. Similarities to the graph of Differences from the graph of y = 2 y = 2 y = 5 2 y = ( + 5) 2 y = (5) 2 y = Mathematics Vision Project MVP Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license
2 4 4. Optima decides to test her ideas using technology. She thinks that it is always a good idea to start simple, so she decides to go with y = She graphs it along with y = 2 in the same window. Test it yourself and describe what you find. 5. Knowing that things make a lot more sense with more representations, Optima tries a few more eamples like y = and y = 2 3, looking at both a table and a graph for each. What conclusion would you draw about the effect of adding or subtracting a number to y = 2? Carefully record the tables and graphs of these eamples in your notebook and eplain why your conclusion would be true for any value of k, given, y = 2 + k. 6. After her amazing success with addition in the last problem, Optima decides to look at what happens with addition and subtraction inside the parentheses, or as she says it, adding to the before it gets squared. Using your technology, decide the effect of h in the equations: y = ( + h) 2 and y = ( h) 2. (Choose some specific numbers for h.) Record a few eamples (both tables and graphs) in your notebook and eplain why this effect on the graph occurs. 7. Optima thought that #6 was very tricky and hoped that multiplication was going to be more straightforward. She decides to start simple and multiply by -1, so she begins with 2012 Mathematics Vision Project MVP Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license
3 5 y = 2. Predict what the effect is on the graph and then test it. Why does it have this effect? 8. Optima is encouraged because that one was easy. She decides to end her investigation for the day by determining the effect of a multiplier, a, in the equation: y = a 2. Using both positive and negative numbers, fractions and integers, create at least 4 tables and matching graphs to determine the effect of a multiplier Mathematics Vision Project MVP Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license
4 NAME Structures of Epressions Ready, Set, Go! Ready Topic: Finding key features in the graph of a quadratic equation Make a point on the verte and draw a dotted line for the ais of symmetry. Label the coordinates of the verte and state whether it s a maimum or a minimum. Write the equation for the ais of symmetry
5 NAME Structures of Epressions What connection eists between the coordinates of the verte and the equation of the ais of symmetry? 8. Look back at #6. Try to find a way to find the eact value of the coordinates of the verte. Test your method with each verte in 1-5. Eplain your conjecture. 9. How many -intercepts can a parabola have? 10. Sketch a parabola that has no - intercepts, then eplain what has to happen for a parabola to have no - intercepts. Set Topic: Transformations on quadratics Choose the area model that is the best match for the equation. 11.! ! 13. 4! 14. 4! A. B C
6 NAME Structures of Epressions D. A table of values for f() = 2 is given. Compare the values in the table for g() to those for f(). Identify what stays the same and what changes. Use this information to write the verte form of the equation of g(). Then graph g(). Describe how the graph changed from the graph of f(). Use words such as right, left, up, and down f() = f() 15. g() = g()
7 NAME Structures of Epressions g() = g() g() = g() g() = g() Go Topic: Finding square roots Simplify a! b! ! ! ! ! 24. 9m! 2p! q!
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