Math 2 Lesson 2-2: Rollercoaster Quadratics Name______________________________ Date _____________________________ Learning Goals: I can predict whether a quadratic will have a minimum or a maximum based on the value of a. I can understand and explain when a quadratic function is increasing and decreasing. I can identify important characteristics of a parabola (y-intercept, max/min, x-intercept, reflective symmetry, etc.) in a table or graph (extend to a problem situations). I can use inequality notation to determine when a quadratic function is above or below a certain value (including using technology) If you love rollercoasters, thank parabolas! Parabolic hills provide great airtime. Floating over maximums and dropping down into the minimums are what coaster enthusiasts crave! Below is a parabolic hill from the Raging Bull coaster located at Six Flags, Illinois. The following equation represents the hill: Left support column y .014 x 2 2.8x 160 Right support column 1. a. The support column furthest to the left is ________ feet tall. The ordered pair for this point is ____________and its mathematical name is ________________________. Describe where this is located in the equation of the parabola: b. The shortest distance from the track to the ground is ________ feet and it occurs __________ feet from the base of each support column. The ordered pair for this point is ______________ and the mathematical name is _____________________. c. As the distance from the left support column increases, the height of the track ___________________ at a(n) ___________________ rate, until you reach the ________________ point of (_______,_______). Then the height of the track will begin to ___________________ at a(n) ___________________ rate. d. What x-value goes with the second occurrence of a height of 160 feet? How did you determine that value? 2. Go back to the graph of the rollercoaster hill. a. Use your calculator to find the x-values for when the track is at least 100 feet above the ground. Lightly shade the part of the graph that represents these intervals. Also describe the intervals of the track that you shaded using the compound inequalities below. ________ < x < ________ ________ < x < ________ b. Use your calculator to find the x-values for when the track is at most 50 feet above the ground. Shade this part of the graph also. Describe the interval by completing the compound inequality below. ________ < x < ________ c. Below is a table of values representing some ordered pairs that make up the parabola. x (feet) 0 40 60 80 100 140 160 180 200 y (feet) 160 109.6 42.4 25.6 20 25.6 70.4 109.6 Underline the input/output pair of numbers that represent the y-intercept of the function. From the table, how do you know this is the y-intercept? Circle the input/output pair of numbers that represent the minimum of the function. From the table, how do you know this is the minimum? There are some values in the table that have not been filled in. Use the symmetric nature of the parabola to fill in each of those values. ***Verify your intervals from 2a & 2b by making a table in the calculator. Also, confirm the values you filled in the table To make a table: / T b 2Table 5 Edit Table Settings . . . 3. Recall that the height function for dropping pumpkins was h(t ) 144 16t 2 and for a basketball long shot was h(t ) 8 40.8t 16t 2 . Describe the similarities between the graphs of the rollercoaster hill and those of the pumpkin drop and basketball functions. Describe the differences between the graphs of the rollercoaster hill and those of the pumpkin drop and basketball functions. ***How is the rule for the rollercoaster hill function similar to and how is it different from the rules of the pumpkin drop and basketball functions? PRACTICE 1. Below is a picture of the third hill (taller one) of the Intimidator rollercoaster at Carowinds amusement park, located in North Carolina. Suppose the following equation represents the graph of the hill, where x represents the horizontal distance in feet and y represents the vertical distance in feet. y .00832 x 2 2.08 x 20 a. Use your calculator to complete the table of values and then sketch a graph the function. x (feet) y (feet) 0 25 50 75 100 125 150 175 200 225 250 b. What are the coordinates of the y-intercept? _________ How do you know from the table? From the equation? c. What are the coordinates of the point where the tallest support column meets the track? __________ What is the mathematical name of this point? ____________________________ d. Use a compound inequality to describe the x-values where the track is at least 144.8 feet above the ground. e. Use compound inequalities to describe the x-values where the track is at most 103.2 feet above the ground. Math 2 2-2 Practice Name_____________________________________ ***NO CALCULATOR*** y 2 x 2 12 x 12 2. Answer the following questions about the equation: a. Is this function linear or non-linear? How do you know? b. Does this parabola have a maximum or a minimum (which one)? How do you know? c. What is the y-intecept of this function? How do you know? d. Complete the table of values. x y 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 -12 -6.5 1.5 5.5 6 4 1.5 -2 e. Underline the input/output pair of numbers that represent the y-intercept of the function. f. Circle the input/output pair of numbers that represent the maximum of the function. g. When does the function have a y-value of at most 1.5? Your answer should be a compound inequality. h. When does the function have a y-value of at least 4? i. Your answer should be a compound inequality. j. Describe the behavior of the function as it approaches its maximum/minimum (whichever applies). Your answer should address both the direction of the function and its rate of change. k. Describe the behavior of the function after its maximum/minimum (whichever applies). Your answer should address both the direction of the function and its rate of change.