During a mission aboard a research station, Emek and Tara responded to unexpected fluctuations in the quantum reactor. The diagnostic system displayed several interconnected function graphs alongside their equations, each describing a different part of the reactor's behavior. By comparing the equations with the graphs, they identified which functions represented stable operating conditions and which showed rapid changes that could lead to instability. They verified their conclusions by examining intercepts, turning points, and intervals where the functions increased or decreased. After selecting the correct operating model, they recalibrated the reactor, restored stable power to the station, and prevented a complete system shutdown.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Use vertex form and examine the sign of the squared term.
A reactor model is given by f(x) = (x - 3)^2 - 4. A technician claims that its graph has a maximum at (3, -4). Which statement correctly checks the claim?
The claim is incorrect because the graph opens upward and has a minimum at (3, -4).The claim is correct because every quadratic graph has a maximum.The claim is incorrect because the turning point is (-3, 4).The claim is correct because the graph opens downward.Correct! The positive coefficient makes the parabola open upward, so its vertex is a minimum.
Check both the vertex and the direction in which the parabola opens.
Next🎉 Congrats! You got this question!
Hint
For ax^2 + bx + c, calculate x = -b/(2a).
The reactor output is modeled by g(x) = -x^2 + 8x - 7. At what integer value of x does the output reach its turning point?
Correct! The turning point occurs at x = 4.
Use the coefficients of x² and x in the axis-of-symmetry formula.
Next🎉 Congrats! You got this question!
Hint
Use the intercepts to form the factors, then choose the sign that gives a minimum.
A stable reactor model must cross the x-axis at x = -2 and x = 5, decrease until its turning point, and then increase. Which equation matches all of these conditions?
y = (x + 2)(x - 5)y = -(x + 2)(x - 5)y = (x - 2)(x + 5)y = (x + 2)(x + 5)Correct! The factors give the required intercepts, and the positive leading coefficient makes the graph decrease and then increase.
Check both the zeros of the function and whether the parabola opens upward or downward.
Next🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!
