choosing STAT #1: Edit. Next, press STAT again and arrow right to CALC then choose #5 Quad Reg. Detailed regression directions and directions for the TI-89 are in the appendix at the back of this packet. stimuli seconds _ _ _ _ _ Record the Quadratic Regression Equation: Important note about scientific notation: When the TI graphing calculator has a very small number, it will convert that number to scientific notation. When writing your equation, it will be helpful for you to translate the number to standard notation. For example, the number 3.0906813 E -5 = 0300030906813. 2. b. Rex is a border collie who is 62 months old. Use the regression equation you found in part a to predict the time it will take Rex to respond to stimuli. d. Explain what the vertex means in this situation. e. Does it make sense that this function has no x-intercepts? Explain. 2. We've seen quadratic models for the museum admission price and the amaranth plants, what happens when we want to model real data with a quadratic function? Recall in Units 2 and 4 that we used regression to find best-fit linear and exponential functions. Similarly, we can use quadratic regression to find a best-t quadratic function. In 2014, cognitive science researchers studied 145 border collies (6 months to 14 years) with humans and objects or food as attention attractors, in order to assess the dogs' attention capture, sustained and selective attention. and sensorimotor abilities. The study results reveal differences in task relevance in sustained attention performance when watching a human or a moving object, which may be explained by life-long learning processes involving such stimuli. The gure at right shows still frame video from an experiment in the study. During task switching researchers found that dogs' selective attention and sensorimotor abilities showed differences between age groups. with performance peaking at middle age. Dogs' sensorimotor abilities showed a quadratic distribution with age. Research results were published in the February 7, 2014 issue of Frontiers in Psychology. a. The table below gives some of the data from the study {it would take a long time to enter all 145 data points into your calculator). Use this data to find a quadratic regression equation and create a detailed sketch of the graph. Recall that you will first enter the data into L1 and L2 on the Tl-83/84 calculator, by Learning outcomes: 1. Communication: explain the meaning of function values 2. Multiple Representations: work with the equation and graph of a quadratic function to identify important aspects of a situation. Use technology to analyze graphs of quadratic functions. 3. Skills and applications: Use the equation of a function to find values that have meaning in context. Warm up: skills problems Simplify the following by using the distributive property: {3} (x - 3)(x + 2) {bl (4 - 5x)(2 x) (C) (3x - 2)2 (d) 4x(2x 7) (e) 5x (x2 + 13x 12) (f) (x2 + (5)0:2 6) Class Discussion: A museum currently charges 56 admission, and on an average weekend 200 people visit the museum. The manager of the museum's business operations would like to increase the price of admissions, but in researching this option, she has found that for every dollar increase in price, she will lose 15 customers. The manager knows that she can determine the weekend revenue by multiplying the number of visitors by the ticket price, so she writes an equation to model revenue. She simplifies the equation, resulting in the function r(x) = 45):2 + 110x +1200 , where x is the number of dollars the price would be increased, and r(x) is the weekend revenue. 1. Describe the shape of the graph. How is it different than the shape of linear or exponential functions? 2. What does the graph tell you about what happens as the price is raised? 3. Use the given information above to find the current weekend revenue. What point on the graph represents this information? How can we verify the point using the equation? 4. Mark the graph and estimate the value of the xintercept on the left side of the graph. What does this point mean in this situation? 5. Mark the graph and estimate the value of the x-intercept on the right side of the graph. What does this point mean in this situation? Group problems: 1. Amaranth is a plant that is grown around the world, and while some species are considered weeds. others contain leaves and seeds that people eat. When amaranth plants are grown in rows, the height that the plants attain is a quadratic function of the spacing between plants within a row. The function H(x) = C'i.0237"x2 1.2780c + 33.2544 models the height (in cm) of plants as a function of spacing between plants (in cm). Rows ofpicm ts a. Estimate the y-intercept on the graph and then find a more precise estimate using your calculator. b. What is the meaning of the point you found in question 1? c. Estimate the vertex of the graph and then find a more precise estimate using your calculator. 7. Find another point on the graph that shares the same y-coordinate with the y-intercept. a. What is the approximate int-coordinate? What does this value mean in the situation? b. We can verify our estimate using a graphing calculator: return to the graph of r(x) and find where it intersects the line y = 1200. How close was your estimate? 8. Mark the graph and estimate the highest point on the graph, called the vertex. a. What are your estimated coordinates? b. What does this tell you about this situation? c. Use your calculator's graphing utility to find a precise vertex. What values does the calculator give? How close was the estimate? e. The graph at right includes data points for all 145 border collies in the research study and the quadratic regression equation the researchers used. Note: the y-axis has scale 0.25, but the computer program researchers used rounded to the tenth place, so the y- scale looks to be jumping oddly. Use the same window to graph the regression equation you found in part a. How does your graph compare to the researchers' graph? 0 20 40 60 BO 100 120 140 160 Age In months d. Use the capabilities of your graphing calculator to find the vertex of the quadratic regression equation from part a. For Tl-83/84 users, this is 2rld CALC #3: Minimum. Record the vertex in the space below. Does it match the researchers' graph above