While guiding a spacecraft through a crowded orbital corridor, Emek and Tara monitored several candidate flight paths displayed as function graphs on the navigation console. Each graph represented a different guidance algorithm with distinct intercepts, turning points, and intervals where the path approached nearby satellites. They compared these key properties to determine which trajectory maintained safe separation while still reaching the destination efficiently. After confirming their choice with the onboard navigation system, they followed the selected path through the busy corridor. The spacecraft arrived at its destination without incident, proving that careful analysis of function graphs was essential for safe navigation.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Find the vertex using x = -b/(2a), then substitute that value into the function.
One candidate flight path is modeled by f(x) = x² - 6x + 5. At which point does the graph reach its turning point?
(3, -4)(-3, 4)(6, 5)(1, 0)Correct! The graph has its turning point at (3, -4).
Find the x-coordinate of the vertex first, then calculate the corresponding y-value.
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Hint
Set g(x) equal to 0 and determine how many distinct real solutions the equation has.
Another flight path is modeled by g(x) = 2x² - 8x + 6. How many x-intercepts does its graph have?
Correct! The graph has 2 distinct x-intercepts.
Factor the quadratic or examine its discriminant to count the real zeros.
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Hint
Choose a test value between 2 and 4 and determine the sign of each function.
The navigation system requires a path that stays above the x-axis between x = 2 and x = 4. Path A is f(x) = (x - 1)(x - 5), and Path B is h(x) = -(x - 1)(x - 5). Which path meets the requirement on the interval 2 < x < 4?
Path A onlyPath B onlyBoth pathsNeither pathCorrect! Path B is positive throughout the interval 2 < x < 4.
Test a value such as x = 3 in both functions and compare their signs.
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