As their spacecraft approached an unexplored star system, Emek and Tara discovered that the navigation console had malfunctioned, separating flight equations from their corresponding trajectory graphs. To establish a stable orbit around the target planet, they had to match each equation with the graph that showed its key features, including intercepts, turning points, and overall shape. By carefully interpreting how each function behaved, they identified the correct trajectory and rejected misleading alternatives. The restored guidance system locked onto the proper orbital path, allowing the spacecraft to enter orbit safely and begin its scientific mission.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Compare the equation with the vertex form f(x) = a(x - h)^2 + k.
The navigation console displays the function f(x) = (x - 2)^2 - 3. Which description matches its trajectory graph?
An upward-opening parabola with vertex (2, -3)A downward-opening parabola with vertex (2, -3)An upward-opening parabola with vertex (-2, 3)A straight line passing through (2, -3)Correct! The positive squared term makes the parabola open upward, and its vertex is (2, -3).
Identify the vertex from the equation and check the sign of the squared term.
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Hint
For ax^2 + bx + c, use x = -b/(2a).
A possible orbital trajectory is modeled by g(x) = x^2 - 6x + 5. What is the x-coordinate of the graph's turning point?
Correct! The turning point occurs at x = 3.
Use the coefficients of x^2 and x to calculate the axis of symmetry.
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Hint
An x-intercept r gives a factor of (x - r), and a positive leading coefficient makes the graph open upward.
The stable trajectory graph must cross the x-axis at x = -1 and x = 4 and open upward. Which equation should Emek and Tara select?
y = (x + 1)(x - 4)y = -(x + 1)(x - 4)y = (x - 1)(x + 4)y = (x + 1)(x + 4)Correct! The factors give x-intercepts -1 and 4, and the positive leading coefficient makes the parabola open upward.
Check both the zeros represented by the factors and the sign of the leading coefficient.
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