A network of deep-space observatories began sending conflicting predictions about an approaching comet. In the mission control center, Emek and Tara examined several function graphs displayed across a digital control wall, each representing a different mathematical model. They compared key features such as intercepts, turning points, intervals of increase and decrease, and end behavior with the latest observational data. One graph consistently matched the comet's measured positions, while the others showed incompatible behavior. After selecting the correct model, they synchronized the observatories, enabling the entire network to coordinate accurate observations as the comet passed through the star system.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Use the vertex form y = a(x - h)^2 + k.
One comet model has equation f(x) = -(x - 1)^2 + 9. Which description matches the graph that Emek and Tara should identify?
An upward-opening parabola with turning point (1, 9)A downward-opening parabola with turning point (1, 9)A downward-opening parabola with turning point (-1, 9)A straight line passing through (1, 9)Correct! The negative leading coefficient makes the parabola open downward, and the vertex is (1, 9).
Identify both the vertex and whether the coefficient of the squared term is positive or negative.
Next🎉 Congrats! You got this question!
Hint
Use the formula x = -b/(2a).
A comet model is given by g(x) = x^2 + 8x + 7. What is the x-coordinate of its turning point?
Correct! The turning point occurs at x = -4.
Substitute the coefficients of x² and x into the vertex formula.
Next🎉 Congrats! You got this question!
Hint
Think about which graph has a minimum at x = 2.
The observatory data show that the comet model should decrease until x = 2 and then increase afterward. Which function best matches this behavior?
y = -x + 2y = -(x - 2)^2 + 1y = x + 2y = (x - 2)^2 + 1Correct! An upward-opening parabola decreases to its vertex at x = 2 and then increases.
Check whether the graph has a minimum or maximum at x = 2 and how it behaves on either side.
Next🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!




