At a deep space observatory, Emek and Tara investigated unexpected telescope readings after the tracking system began pointing at the wrong region of the sky. The computer displayed several function graphs representing possible motion models for the observed object. They compared the graphs' intercepts, turning points, and intervals of increase and decrease with the telescope's recorded observations. One graph clearly contradicted the measured behavior, revealing the source of the tracking error. After selecting the correct model and recalibrating the system, the observatory locked onto its target again, allowing astronomers to continue their research without interruption.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
The graph needs a maximum turning point at t = 3.
The telescope records show that the object's position increases until time t = 3, then decreases. Which motion model has a graph that matches this behavior?
f(t) = (t - 3)² + 2f(t) = -(t - 3)² + 2f(t) = 3t + 2f(t) = -3t + 2Correct! This model increases until t = 3 and decreases afterward.
Look for a downward-opening graph with its turning point at t = 3.
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Hint
Use t = -b/(2a) to find the horizontal coordinate of the vertex.
One possible motion model is g(t) = t² - 8t + 12. At what integer value of t does the graph reach its turning point?
Correct! The graph reaches its turning point at t = 4.
Identify a and b, then apply the vertex formula carefully.
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Hint
First identify the model with zeros at 2 and 6, then test a value between them.
The recorded object crossed the reference axis at t = 2 and t = 6 and was above the axis between those times. Which model is consistent with these observations?
h(t) = (t - 2)(t - 6)h(t) = -(t - 2)(t - 6)h(t) = (t + 2)(t + 6)h(t) = -(t + 2)(t + 6)Correct! The model has the required intercepts and is positive between them.
Check both the intercepts and the sign of the function between t = 2 and t = 6.
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