1. The team has 3 seconds left in the game at their own 38 yard line, facing a decision to kick a 55 yard field goal or attempt a Hail Mary pass.
2. They model the kicker's kick using the equation y = -1/14x^2 + 23/7x – 45/14 and graph it to determine the kick's trajectory.
3. By analyzing the graph, they find that the kick would land 1 yard short of the goal posts, so attempting the long field goal would likely fail.
2. With three seconds left on the clock of our football
game, we are held at the 38 yard line. We can
either wait and take a 55 yard kick, or make a
Hail Mary pass and hope to gain some yardage.
We decide to do some quick thinking and use the
equation y = -1/14x^2 + 23/7x – 45/14 to
represent our kicker’s best kick. Once we graph
our equation, it will tell us where the ball starts
on the field, where it will land, how high it will
go, the path it will take, and
consequently, whether we will make the field
goal or not.
3. To start off, we need to solve our equation for the
vertex which can tell us how high the ball will
go, the vertex on our graph, and the axis of
symmetry. We first use the equation x = -b/2a to
find the x value of our vertex, or the highest
point, of our kick.
y = -1/14x^2 + 23/7x – 45/14 1. Plug in the variables
x = -b/2a
2. Solve
x = (-23/7)/2(-1/14)
x = -3.428571429/- 3. Plug x into the equation
0.1428571429 4. Solve for y
x = 23
y = -1/14(23)^2 + 23/7(23) –
45/14
y = 34.57142857
So our vertex is (23, 34.57142857), the graph has a
maximum of 34.57142857, and our axis of
symmetry is 23. The axis of symmetry is the
turning point of the ball in the air.
4. To find out where the ball will start and land on
the field, we need to factor our equation.
y = -1/14x^2 + 23/7x – 1. Set the equation equal
45/14 to zero
0 = -1/14x^2 + 23/7x – 2. Get rid of the negative
45/14 a and the coefficient of
-14(0 = -1/14x^2 + 23/7x – x^2
45/14) 3. Factor the equation
0 = x^2 – 46x + 45 4. Set the x values equal
0 = (x – 45)(x – 1) to zero and solve for x
0 = x – 45 0=x–1
x = 45 x=1
Now we know that the ball starts at the 45 yard
line and lands 1 yard short of the field goal.
5. Now that we have our vertex and intercepts, we can find other
points on our graph.
1. Start by making a table with x and y values.
2. Have two numbers both above and below the x value of the vertex
3. Plug in two of the x values into your original equation. (make sure that
these values are either one unit above or below the x value, and two units
above or below the x value; never two values the same amount away from
the vertex)
4. Solve for y with each different value
5. Plug what you get for y into the spot on your table that corresponds with
the x value you plugged in
6. Now you have points that lay on your kick’s path to plot on your graph.
X Y
21 34.28571429
22 34.5
23 34.57142857
24 34.5
25 34.28571429
6. Now that we have all of the information that we need, we can graph it. We start
by first plotting the points, and then connecting them. On our graph we have
the axis of symmetry running through the vertex at (23, 34.57142857).
Vertex
Maximum
Axis of symmetry
Roots
•The y-axis represents
the position of the field
goal along the x-axis
(so at x = 0 is the field
goal)
•From the 10 yard mark
to the 0 mark on the x-
axis of the graph is the
end zone
7. To get a more precise graph, we plugged our
information into the graphing calculator.
8. The kicker starts at 45 yards on the opposing
team’s side. He kicks the ball, which peaks at
34.57142857 feet in the air, at 23 yards on
the field. From there it goes back down and
lands 9 yards into the end zone, or 1 yard
away from the field goal. Even if the kicker
were to get that one last yard, his kick would
not have had enough height to make the field
goal.
9. Based off of our graph, we would make a Hail Mary
pass in the last three seconds of the game in
hope of gaining yards. We would not take the
kick at the 38 yard line because our kicker’s kick
would land 1 yard short of the field goal. To
make the field goal, we would need to gain at
least 14 yards. This is because on our graph, the
y-intercept is -3.214285714, and as the field
goal is 10 feet tall, we would need to gain 14
yards. That extra yardage would be the minimum
amount of yards that we would need to gain to
get the kick’s height high enough to get the field
goal. Thus, we will make a Hail Mary pass in
hopes of gaining at least 14 yards.
