A. Lesson Context. B. Lesson Objectives. C. Fast Five (Skills Review Focus)
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1 A. Lesson Context BIG PICTURE of this UNIT: How & why do we build NEW knowledge in Mathematics? What NEW IDEAS & NEW CONCEPTS can we now explore with specific references to QUADRATIC FUNCTIONS? How can we extend our knowledge of FUNCTIONS, given our BASIC understanding of Functions? CONTEXT of this LESSON: Where we ve been In Lessons 2 & 3, you worked with quadratic graphs and equations using the form of y = a(x h) 2 + k Where we are HOW do we apply the vertex form of quadratic models in contextual problems Where we are heading How do we extend our knowledge & skills of quadratic functions, given the new ideas & concepts we now know about functions. B. Lesson Objectives a. Apply the equation y = a(x h) 2 + k to geometrical contexts and to modeling contexts as well as data contexts (scatter plots) C. Fast Five (Skills Review Focus) Solve the equation 0 = - 2(x + 2) Find the vertex of the parabola y = x 2 + 6x 7 A parabola with the equation of y = a(x h) 2 2 goes through the points (1,2) and (- 5,- 1). Find the values of a and h.
2 D. Modeling with Quadratic Functions (Example #1 Geometric Context) A quadratic function is defined by the equation f(x) = x 2 4x 5. (a) Determine the equation of the axes of symmetry. (b) Determine the vertex of this parabola. (c) Rewrite the equation in vertex form and state the optimal value of the quadratic function. (d) Find the zeroes of the parabola. (e) Sketch the parabola, labelling the vertex and the y- intercept. (f) Solve f(x) = - 5. (g) Using your answer from Q(c), write the equation of the INVERSE RELATION of f(x).
3 E. Modeling with Quadratic Functions (Example #2) Mr S s sister is a motorcycle instructor and runs a training school. Because she works for herself, she can charge any amount (as an hourly charge) that she wishes. She keeps track of her hourly fees and her profits and has prepared a DATA TABLE showing the relationship between her hourly wages and her profits. Here is her data set Hourly fee profit In this relation, the variables: X represents the hourly fee my sister charges Y represents the monthly profit she makes (a) Graph the data points (b) Draw the parabola that best fits the data set as well as you can. (c) Determine the equation for this relation. Show your work. (d) My sister would like to know what hourly fee OPTIMIZES her profits? (e) From your equation in question (c), expand and rewrite on standard form (f) Use your TI- 84 to determine the equation of the regression curve that best fits your data set.
4 F. Modeling with Quadratic Functions (Example #3 Modeling Businesses)
5 G. Modeling with Quadratic Functions (Examples #4 Modeling Bridges & Rockets)
6 H. Modeling with Quadratic Functions (Examples #5 Modeling Bridges & Rockets)
7 I. Modeling with Quadratic Functions (Example #6 Modeling Bridges) The underside of a concrete railway underpass forms a parabolic arch. The arch is 24 m wide at the base and 8.0 high in the center. The upper surface of the underpass is 40 m wide. The concrete is 2 m thick at the center. Can a truck that is 5 m wide and 7.5 tall get through this underpass if the time is 3:30pm? (a) Visualize the information by drawing a diagram wherein you label the relevant information. (b) Recall that our general starting point would be a quadratic model and that the vertex form of the equation is y = a(x h) 2 + k (c) How can you use the quadratic equation to address the problem of the truck passing through the underpass?
8 J. Modeling with Quadratic Functions (Example #7) The Next Cup coffee shop sells a special blend of coffee for $2.60 per mug. The shop sells about 200 mugs per day at this price. Customer surveys show that for every $0.05 decrease in price, the shop would sell 10 more mugs per day. (HINT: start with numbers and a data table to see what may be going on) (d) Determine the REVENUE that the coffee shop makes initially, given the price per mug and the amount of mugs sold. (e) Since we are making changes in the pricing & revenues of the coffee shop, we need to decide upon an INDEPENDENT variable to use in modeling a change in the revenues so we need an R(x) equation.. (f) Determine the MAXIMUM daily revenue from coffee sales and the price per mug in order to earn this revenue. (g) Write an equation in both standard form and vertex form to model this problem. Then sketch the graph.
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