Flight Path Worksheet
Flight Path Analysis
Mission: Identifying Parabolic Parameters
Pilot:
Date:
Analyze the following flight paths (parabolas) to determine their critical coordinates and values. Circle the best answer for each mission objective.
1
An aircraft follows the path shown in the radar scan below. What are the coordinates of the vertex?
Center is (0,0)
A) (0, -3)
B) (2, -3)
C) (-3, 2)
D) (2, 0)
2
At which x-coordinates does the flight path intersect the landing strip (x-axis)?
A) x = -2 and x = 2
B) x = 0 and x = 4
C) x = -4 and x = 4
D) x = 2 and x = 5
3
A drone is programmed with the flight path equation: \(y = 3(x - 5)^2 + 12\).
Identify the axis of symmetry for this path.
A) x = -5
B) y = 12
C) x = 5
D) x = 12
4
A satellite's orbit is modeled by \(y = -2(x + 1)^2 - 8\).
Does this path have a maximum or minimum value, and what is it?
A) Minimum of -8
B) Maximum of -8
C) Minimum of -1
D) Maximum of -1
5
The fuel efficiency curve of a jet is given by \(y = x^2 - 6x + 10\).
What is the vertex of this parabola?
A) (3, 1)
B) (-3, 37)
C) (6, 10)
D) (3, 10)
6
A rocket's height over time is modeled by \(h = t^2 - 8t + 12\).
At what times (\(t\)) is the height equal to zero?
A) t = 2 and t = 6
B) t = -2 and t = -6
C) t = 3 and t = 4
D) t = 0 and t = 12
7
Examine the flight log graph. What is the equation for the axis of symmetry?
A) x = 0
B) y = 2
C) x = 2
D) x = -2
8
A glider reaches its lowest point at the vertex of \(y = (x - 4)^2 + 15\).
What is the **minimum height** reached by the glider?
A) 4 units
B) -4 units
C) 15 units
D) 0 units
9
If a parabola has a maximum value, which of the following must be true about its equation \(y = ax^2 + bx + c\)?
A) a > 0
B) a < 0
C) c = 0
D) a = 1
10
A projectile is launched such that its path is \(y = -x^2 + 10x\).
What are the zeros (landing points) of this trajectory?
A) x = 0 and x = 5
B) x = 0 and x = 10
C) x = 5 and x = 10
D) x = -10 and x = 10
Unit: Quadratic Systems Log ID: P-404-B Status: Field Test
Flight Path Answer Key
Answer Key
Flight Path Analysis: Mission Debrief
Control Center Use Only
1
Answer: B — (2, -3)
The vertex is the lowest point on this parabola. Counting from the center (0,0), the point is 2 units right and 3 units down.
2
Answer: A — x = -2 and x = 2
The zeros (x-intercepts) are the points where the curve crosses the horizontal axis (landing strip).
3
Answer: C — x = 5
In vertex form \(y = a(x - h)^2 + k\), the axis of symmetry is always the vertical line \(x = h\). Since the equation has \((x - 5)\), \(h = 5\).
4
Answer: B — Maximum of -8
The leading coefficient \(a = -2\) is negative, meaning the parabola opens downward and has a maximum. The value is the k-coordinate of the vertex (\(k = -8\)).
5
Answer: A — (3, 1)
Using \(x = -b / 2a\): \(x = -(-6) / 2(1) = 3\). Substitute \(x = 3\) back in: \(3^2 - 6(3) + 10 = 9 - 18 + 10 = 1\).
6
Answer: A — t = 2 and t = 6
Factor the quadratic: \(t^2 - 8t + 12 = (t - 2)(t - 6)\). Setting factors to zero gives \(t = 2\) and \(t = 6\).
7
Answer: C — x = 2
The axis of symmetry is the imaginary vertical line (dashed red in the graph) that splits the parabola into two mirrored halves. It passes through \(x = 2\).
8
Answer: C — 15 units
The minimum height occurs at the vertex. For \(y = (x - 4)^2 + 15\), the vertex is (4, 15). The height is the y-value: 15.
9
Answer: B — a < 0
A parabola has a maximum value only if it opens downward. This occurs when the leading coefficient \(a\) is negative.
10
Answer: B — x = 0 and x = 10
Solve \(-x^2 + 10x = 0\) by factoring out \(-x\): \(-x(x - 10) = 0\). This gives solutions \(x = 0\) and \(x = 10\).
End of Key — Confidential Reference
