Name: ______Date: ______Hr: ______
Trigonometric Models Activity #1
Motion Graphs and Trigonometric Functions
Complete the first set of questions as a class, using a CBR hooked into the overhead graphing calculator.
PART I: Constant Speed:
1.Have a student walk SLOWLY and STEADILY away from the sonic ranger. Sketch the graph they make at right.
(a)The D(M) on the axes stands for: ______
(b)The T(S) on the axes stands for: ______
(c)What location has a D-value of zero?
(d)What time has a T-value of zero?
2.Now have a student walk QUICKLY and STEADILY away from the sonic ranger. Sketch the graph they make at right.
(a)What is the difference between this graph and the graph from question #1?
(b)What would the graph look like if a 3student walked away from the sonic ranger faster than student 1 and slower than student 2? Describe it.
3.Have a student walk SLOWLY and STEADILY towards the sonic ranger. Sketch the graph they make.
(a)What is the difference between this graph and the graphs from questions #1 and 2?
(b)What would the graph look like if a student walked QUICKLY and STEADILY towards the sonic ranger? Describe it.
4.Draw a prediction of the graph you think will occur if a student follows the motion described:
▪student walks SLOWLY and STEADILY away from the sonic ranger for 5 seconds
▪student STOPS and stands still for 5 seconds
▪student walks QUICKLY towards the sonic ranger for 3 seconds
▪student STOPS and stands still for the last 2 seconds
Have someone in the class walk the motion as described. Did your predicted graph match? Explain.
5.Questions #1-4 are all composed of segments of constant speed. What kind of graph results from this constant speed motion? What kind of function could be used to model a distance vs. time relationship for an object going at a constant speed?
Part II: Speeding Up and Slowing Down
6.Roll a ball or cart down a ramp towards the sonic ranger, which is placed at the bottom of the ramp. Sketch the resulting graph:
(a)What is the difference between this graph and those in question 1-4?
(b)On this graph, what location has a D-value of zero?
7.Push a ball or cart up a ramp away from the sonic ranger, which is placed at the bottom of the ramp. Allow it to roll up the ramp and back down. Sketch the resulting graph:
(a)What is the difference between this graph and that in question 6?
(b)On this graph, what location has a D-value of zero?
8.Now move the sonic ranger so it is placed at the TOP of the ramp. PREDICT what the graph will look like if you push a ball or cart up the ramp towards the sonic ranger, allowing it to roll up the ramp and back down.
(a)Have someone roll the ball as described. Does the graph match your prediction? Explain.
(b)What is the difference between this graph and that in question 7?
(c)On this graph, what location has a D-value of zero?
9.PREDICT what the graph will look like if you hold a motion detector up high and allow a ball to bounce underneath it. Have someone bounce the ball as described.
(a)Does the graph match your prediction? Explain.
(b)On this graph, what location has a D-value of zero?
10.Repeat the ball bouncing, but this time use “BALL BOUNCE” from the APPLICATIONS menu. Sketch the resulting graph.
(a)How is this graph different than the graph generated in question 9?
(b)On this graph, what location has a D-value of zero?
11.Questions #5-10 are all composed of segments of speeding up or slowing down. What kind of graph results from this changing speed motion? What kind of function could be used to model a distance vs. time relationship for an object that is speeding up or slowing down?
Part III: Periodic Motion
12.Release a pendulum in front of the sonic ranger and let it swing back and forth. Sketch the resulting graph.
(a)How is this graph different from the ball bouncing graph?
(b)Use the TRACE function on your calculator to estimate how long it takes the pendulum to swing back and forth through one complete cycle.
13.How would the pendulum graph be different if you released the pendulum from a HIGHER location than before? Try it and sketch the resulting graph.
(a)What is the difference between this graph and #12?
(b)Use the TRACE function on your calculator to estimate how long it takes the pendulum to swing back and forth through one complete cycle. Is this about the same or much different than the previous graph from #12?
14.Questions #12-13 are composed of segments which repeat themselves over and over. What kind of function could be used to model a distance vs. time relationship for an object that is repeating a cycle over and over?
15.Think of three other things in the natural world that follow repeating cycles. List them below.
Part IV: Samples of Periodic Functions:
Complete the table of values and then use the axes provided to graph each formula over the domain specified. Add units to both the x- and y-axes.
16.y = sin xwhere 0° x 360°
x° / y / x° / y0 / 0 / 210
30 / 0.5 / 225
45 / 0.7 / 240
60 / 0.9 / 270
90 / 1 / 300
120 / 315
135 / 330
150 / 360
180
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