4.1+Tracking+a+Ball

February 9, 2009 JSM  Big Idea: Quadratic functions can help us describe situations in the real world. Essential Question: How can I make a prediction using quadratic relationships? __Notes__: We worked on Problem 3.1, 3.2, Mathematical Reflections 3, and Problem 4.1 in class. __Did you know?__ – The objects hit the ground at the same time.  __Constant Term – A term without any variables__.  **__ Investigation 4: What Is a Quadratic Function? __**   **__ 4.1 Tracking a Ball __** <span style="font-family: 맑은 고딕;"> <span style="font-family: 맑은 고딕;">Suppose you throw a ball straight up into the air. This table describes how the height of the ball might change as it travels through the air.
 * <span style="font-family: 맑은 고딕;">Time (s) || <span style="font-family: 맑은 고딕;">Height (ft)  ||
 * <span style="font-family: 맑은 고딕;">0.00 || <span style="font-family: 맑은 고딕;">0  ||
 * <span style="font-family: 맑은 고딕;">0.25 || <span style="font-family: 맑은 고딕;">15  ||
 * <span style="font-family: 맑은 고딕;">0.50 || <span style="font-family: 맑은 고딕;">28  ||
 * <span style="font-family: 맑은 고딕;">0.75 || <span style="font-family: 맑은 고딕;">39  ||
 * <span style="font-family: 맑은 고딕;">1.00 || <span style="font-family: 맑은 고딕;">48  ||
 * <span style="font-family: 맑은 고딕;">1.25 || <span style="font-family: 맑은 고딕;">55  ||
 * <span style="font-family: 맑은 고딕;">1.50 || <span style="font-family: 맑은 고딕;">60  ||
 * <span style="font-family: 맑은 고딕;">1.75 || <span style="font-family: 맑은 고딕;">63  ||
 * <span style="font-family: 맑은 고딕;">2.00 || <span style="font-family: 맑은 고딕;">64  ||
 * <span style="font-family: 맑은 고딕;">2.25 || <span style="font-family: 맑은 고딕;">63  ||
 * <span style="font-family: 맑은 고딕;">2.50 || <span style="font-family: 맑은 고딕;">60  ||
 * <span style="font-family: 맑은 고딕;">2.75 || <span style="font-family: 맑은 고딕;">55  ||
 * <span style="font-family: 맑은 고딕;">3.00 || <span style="font-family: 맑은 고딕;">48  ||
 * <span style="font-family: 맑은 고딕;">3.25 || <span style="font-family: 맑은 고딕;">39  ||
 * <span style="font-family: 맑은 고딕;">3.50 || <span style="font-family: 맑은 고딕;">28  ||
 * <span style="font-family: 맑은 고딕;">3.75 || <span style="font-family: 맑은 고딕;">15  ||
 * <span style="font-family: 맑은 고딕;">4.00 || <span style="font-family: 맑은 고딕;">0  ||

<span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">a. <span style="font-family: 맑은 고딕;">The height of the ball increase from 0 until its maximum point, then decreases until it hits 0. <span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">b. <span style="font-family: 맑은 고딕;">The graph would have a shape of a parabola (upside-down U). It’s range would be 0 to 4. Will have a maximum point of (2, 64), x-intercepts of (0, 0), (4, 0), and y-intercept of (0, 0). <span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">c. <span style="font-family: 맑은 고딕;">These data represent a quadratic function. Because the pattern of a data is exactly the same as the pattern of a data of situations we learned before in Investigation 1. And the shape of the graph is parabola which means that these data represent a quadratic function. <span style="font-family: 맑은 고딕;"> **__ <span style="font-family: 맑은 고딕;">Problem 4.1 Follow-Up __** <span style="font-family: 맑은 고딕;"> <span style="font-family: 맑은 고딕;">b. Yes, this graph matches my description from part B of Problem 4.1. The shape of the graph is identical with my descriptions, the maximum values and the range of the graph was also correct. <span style="font-family: 맑은 고딕;">c. The ball reaches a height of 58 feet on about 1.38 seconds and 2.61 seconds. <span style="font-family: 맑은 고딕;">d. h = -16t2+64t = -40.96+102.4 = 61.44 <span style="font-family: 맑은 고딕;"> <span style="font-family: 맑은 고딕;">b. The constant term can effect the table and the graph by changing the starting value. The information the constant term in this equation gives you that the y-intercept is (0, 6) <span style="font-family: 맑은 고딕;">i. The maximum height reached by the ball is different. The Q2’s maximum height is 6 ft greater than the Q1. <span style="font-family: 맑은 고딕;">ii. The x-intercepts are different. Q1’s x-intercepts are: (0, 0), (4, 0). Q2’s (estimated) x-intercepts are: (0.09, 0), (3.9, 0) <span style="font-family: 맑은 고딕;">  iii. The general patterns are the same. The both values increase until its maximum point, then drops down to 0.
 * <span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">A.   **** <span style="font-family: 맑은 고딕;">Describe how the height of the ball changes over this 4-second time period.  **
 * <span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">B.   **** <span style="font-family: 맑은 고딕;">Without making a graph, describe what the graph of these data would look like. Include as many important features as you can.  **
 * <span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">C.   **** <span style="font-family: 맑은 고딕;">Do you think these data represent a quadratic function? Explain why or why not.  **
 * <span style="font-size: 14pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">1.   **** <span style="font-family: 맑은 고딕;">The height, //h,// of the ball after //t// seconds can be described by the equation //h// = -16//t//2+64//t//.  **
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">a.   **** <span style="font-family: 맑은 고딕;">Graph this equation on your calculator.  **
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">b.   **** <span style="font-family: 맑은 고딕;">Does the graph match your description from the part B of Problem 4.1? Explain.  **
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">c.   **** <span style="font-family: 맑은 고딕;">Use the equation or the graph to figure out when the ball reaches a height of about 58 feet.  **
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">d.   **** <span style="font-family: 맑은 고딕;">Use the equation to find the height of the ball after 1.6 seconds.  **
 * <span style="font-size: 14pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">2.   **** <span style="font-family: 맑은 고딕;">In Problem 4.1, the initial height of the ball is 0 feet. This is not very realistic because it means that you would have to lie on the ground and release the ball without extending your arm. A more realistic equation for the height of the ball after //t// seconds is h=-16t2+64t+6.  **
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">a.   **** <span style="font-family: 맑은 고딕;">Make a table and a graph for this quadratic function.  **
 * <span style="font-family: 맑은 고딕;">Height (ft) || <span style="font-family: 맑은 고딕;">Time (sec)  ||
 * <span style="font-family: 맑은 고딕;">6 || <span style="font-family: 맑은 고딕;">0  ||
 * <span style="font-family: 맑은 고딕;">34 || <span style="font-family: 맑은 고딕;">0.5  ||
 * <span style="font-family: 맑은 고딕;">54 || <span style="font-family: 맑은 고딕;">1  ||
 * <span style="font-family: 맑은 고딕;">66 || <span style="font-family: 맑은 고딕;">1.5  ||
 * <span style="font-family: 맑은 고딕;">70 || <span style="font-family: 맑은 고딕;">2  ||
 * <span style="font-family: 맑은 고딕;">66 || <span style="font-family: 맑은 고딕;">2.5  ||
 * <span style="font-family: 맑은 고딕;">54 || <span style="font-family: 맑은 고딕;">3  ||
 * <span style="font-family: 맑은 고딕;">34 || <span style="font-family: 맑은 고딕;">3.5  ||
 * <span style="font-family: 맑은 고딕;">6 || <span style="font-family: 맑은 고딕;">4  ||
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">b.   **** <span style="font-family: 맑은 고딕;">The 6 in the equation above is called a constant term. A __constant term__ is a term that does not contain a variable. How does the constant term affect the table and the graph? What information does the constant term in this equation give?  **
 * <span style="font-size: 13pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">c.   **** <span style="font-family: 맑은 고딕;">Compare the graphs of the equations in questions 1 and 2. Discuss the similarities and differences in the following.  **
 * <span style="font-size: 12pt; mso-fareast-font-family: '맑은 고딕'; mso-fareast-theme-font: minor-latin; mso-bidi-font-family: '맑은 고딕'; mso-bidi-theme-font: minor-latin;"> <span style="font-family: 맑은 고딕;">i.   **** <span style="font-family: 맑은 고딕;">the maximum height reached by the ball  **
 * <span style="font-family: 맑은 고딕;">ii. the x-intercepts **
 * <span style="font-family: 맑은 고딕;">iii. the pattern of change in the height of the ball over time **