Journal
2.5.4 Journal: Factoring and Graphing
Name:
Algebra I Sem 2 (S4539132)
____________________
Points possible: 20
Date: ____________
Scenario: The Cannon
Instructions:
●
View the video found on page 1 of this Journal activity.
●
Using the information provided in the video, answer the questions below.
●
Show your work for all calculations.
The Cannon: Ernest’s friend Nik is about to be shot out of a cannon. The path he will travel follows a
parabolic arch that can be described by this polynomial.
2
f(x) = –0.05(x – 26x – 120)
He is supposed to land on a safety net 30 feet away. Does the function give you enough information to tell
you where he will land? If so, how far from the cannon will he land?
Make Sense of the Problem
(3 points: 1 point for each answer)
What do you know?
The trajectory through the air can be defined by a polynomial and
the net where he’s going to lend is 30 feet away from the cannon.
What do you want to
find out?
We want to find out exactly where Nik will land and whether he is
going to land on the net or not.
What kind of
answers do you
expect?
I expect the answer to be a positive number because the distance
can never be negative and also, the answer will be expressed in
feet.
2
than 1. The function f(x) = –0.05(x – 26x – 120) represents the path coming out of the cannon. If x is the
horizontal distance from the cannon, what does f(x) represent? (1 point)
f(x) represents the distance of the cannon from the ground.
OR
f(x) = The vertical distance (height) of the cannonball at point x.
2. What do the zeros of this function represent? Remember the zeros are where f(x) = 0. (1 point)
The zeros are the points where the height, f(x) = 0
Therefore;
The zeros represent the points at which the cannonball is at ground level
3. Will the zeros tell you where the net should be placed? Why or why not? (1 point)
The net should be placed where the cannonball is expected to reach ground level.
Therefore;
●
The zeros indicate where the net should be placed
Yes, it would. By knowing the spot where the cannon will hit the ground, we can set the net at
the spot.
4. To find the zeros, set the function equal to 0, and then factor the polynomial. Start by setting the
function equal to 0. (1 point)
f(x) = 0
-0.05 (x² - 26x -120) = 0
x² - 26x - 120 = 0
Factor the polynomial.
2
2
The polynomial (x – 26x – 120) is in the form x + bx + c which can be factored as (x + p)(x + q) .
2
5. Next factor the polynomial x – 26x – 120. To identify the factors, complete the table: Note: This table
does not contain all the factors of –120, but it has enough to let you factor the polynomial. (2 points: 0.5
points for each row)
p
q
p+q
10
–12
-2
–10
12
2
30
–4
26
4
–30
-26
6. Which values of p and q give the correct factors? (1 point)
The value of p and q that gives the correct factors are -30 and 4 since it gives p+q = -26
7. Factor the polynomial completely: (2 points: 1 point for each factor)
2
0 = –0.05(x – 26x – 120)
-0.05 (x² - 26x - 120) = 0
-0.05 (x+4) (x-30) = 0
(x+4) (x-30) = 0
8. Find the roots of the equation by setting each factor equal to 0 and solving for x.
Hint: There are three factors, but the constant factor, –0.05, does not equal zero. Solve for x with the
other two factors. (2 points: 1 point for each factor)
x+4 = 0 and x-30=0
x=-4 and x=30
9. Identify the roots. (2 points: 1 point for each root)
The roots of the equation is x = -4 and x = 30
Find the zeros from the graph.
2
This is the graph of f(x) = –0.05(x – 26x – 120).
10. What are the zeros of this function? Circle them on the graph. (1 point)
x = -4 and x = 30
11. Are these zeros the same as the roots you found? (1 point)
Yes, they are!
12. What does a negative zero mean in terms of this problem? (1 point)
The zeros of the function are x = 30 and x = -4, which are points at
which the projectile (cannonball) is at ground level. The negative zero, x
= -4, means that, apart from the point x = 30, the projectile can (also/only)
be at ground level at a point before the location where the projectile is
launched, (-4) which means that at the point at which the cannonball is
launched, which is at x = 0, the cannonball was not on the ground, but
at the lip of the barrel or at a more elevated position.
13. Following this trajectory, will Nik hit a net that is 30 feet from the cannon? How do you know? (1
point)
I have 2 answers for this: At 30 feet, x = 30, which is a zero of the function, and the cannonball
is at ground level, such that a net is placed at 30 feet from the cannon
will be hit by the cannonball.
Therefore;
●
Yes Nik will hit a net that is 30 feet from the cannon
OR
The distance between x = -4 and x = 30 is 34 units. If the cannon was fired from the point
when x = -4, the cannon will hit the ground again 34 units from the point it was fired from. If
Nik put a net of 30 units from the firing point, the cannon will fly past it.