Quad Toughies
1. A football player kicks a field goal. After reaching its maximum height, the ball starts to fall, passes
between the goal posts, then hits the ground. When the ball passes between the goal posts, it is at a
height of 3.15m. In the Cartesian plane below, a portion of the parabola represents function f. Function f
represents the height of the ball in relation to the time elapsed since the kick.
The rule of function f is 𝑓(𝑥) = −0.6(𝑥 − 2.5)2 + 3.75.
How much time elapses between the moment the ball passes between the goal posts and the moment
it hits the ground?
2. A rocket was launched during a fireworks show. The side view of the rocket’s parabolic trajectory is
represented by the following table of values and graph.
x
(metres)
y
(metres)
9
19
29
39
54
126
150
126
The rocket exploded directly above the fountain. The distance between the fountain and the launching
point is 40 m. What height (h) was the rocket when it exploded?
3. The side view of the trajectory of a volleyball is represented in the following Cartesian plane.
The trajectory is represented by a parabola whose vertex is (9, 3). The scale of the graph is given in
metres. How far away from the net will the volleyball hit the ground?
4. The stream of water coming out of a fountain lands on a child’s head. The child’s height is 0.9m. The
child wants to move so that the stream of water lands at his feet. The trajectory of the stream of water
is represented in the following Cartesian plane. The scale of the graph is in metres.
The rule 𝑓(𝑥) = −0.1(𝑥 − 2.2)2 + 2.5 represents the trajectory of the stream of water. What distance
must the child so that the water will lad at his feet?
5. Diagonal AC was drawn in rectangle ABCD shown below. In addition:
̅̅̅̅ = (𝑥 − 10)cm
m𝐴𝐷
B
A
̅̅̅̅ = (𝑥 + 11)cm
m𝐷𝐶
̅̅̅̅ = 39cm
m𝐴𝐶
D
What is the numerical perimeter of rectangle ABCD?
C
6. The following graph represents the side view of the path of a dolphin as it performs a trick during a
show at an aquarium. This path is composed of portions of tow parabolas associated with functions f
and g respectively. The scale of the graph is in metres.
5
The rule 𝑓(𝑥) = 9 (𝑥 − 3)2 − 5 represents the dolphin’s path when it is in the water. When it is out of
the water, the dolphin reaches a maximum height of 4 metres. The distance between points A and C is
10 metres. What is the rule of function g?
7. A Tennis player hits a ball against a wall. At the moment the player hits the ball, it is 1 m above the
ground. The ball reaches a maximum height of 3 m. On its way down, the ball hits the wall at a point
2.28 m above the ground. The side view of the ball’s trajectory is illustrated below.
1
The rule representing this trajectory is 𝑓(𝑥) = − 8 (𝑥 − 4)2 + 3.
At the moment the player hits the ball, what is the distance between the ball and the wall?
8. The equation ℎ(𝑥) = −2𝑡 2 + 10𝑡 gives the relationship between the height in metres of a golf ball,
and the time in seconds since it has been hit. How long after being hit will the golf ball be 8 metres?
9. Melanie was playing with a remote-controlled toy airplane. The plane took off from a balcony and
landed on the ground 8 minutes later. Three minutes after taking off, the plane reached a maximum
altitude of 10 metres. In the graph below, the plane’s altitude as a function of time is represented by a
portion of a parabola.
How high off the ground is the balcony located?
10. In the figure on the right, quadrilateral ABCD is a
rectangle, and quadrilateral DCFE is a square.
The area of rectangle ABCD is 240 cm2. In addition:
̅̅̅̅ = (𝑥)cm
m𝐴𝐷
̅̅̅̅ = (2𝑥 + 4)cm
m𝐴𝐵
What is the numerical area of square DCFE in cm2?