Question

1. The function f(x) = –0.05(x2 – 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)

2. What do the zeros of this function represent? Remember the zeros are where f(x) = 0. (1 point)

3. Will the zeros tell you where the net should be placed? Why or why not? (1 point)

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)

Factor the polynomial.

The polynomial (x2 – 26x – 120) is in the form x2 + bx + c which can be factored as (x + p)(x + q) .

5. Next factor the polynomial x2 – 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
–10 12
30 –4
4 –30

6. Which values of p and q give the correct factors? (1 point)

7. Factor the polynomial completely: (2 points: 1 point for each factor)
0 = –0.05(x2 – 26x – 120)

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)

9. Identify the roots. (2 points: 1 point for each root)

10. What are the zeros of this function? Circle them on the graph. (1 point)

11. Are these zeros the same as the roots you found? (1 point)

12. What does a negative zero mean in terms of this problem? (1 point)

13. Following this trajectory, will Nik hit a net that is 30 feet from the cannon? How do you know? (1 point)

Answer 1

Question 1

f(x) represents the distance of the cannon from the ground

Question 2

When f(x) = 0, there will two values of 'x'. The point on the positive x-axis where the graph crosses represent the point where the cannon hits the ground.

Question 3

Yes, it would. By knowing the spot where the cannon will hit the ground, we can set the net at the spot.

Question 4

f(x) = 0
-0.05 (x² - 26x -120) = 0
x² - 26x - 120 = 0

Question 5

Please refer to the table attached below

Question 6

The value of p and q that gives the correct factors are -30 and 4 since it gives p+q = -26

Question 7

Factorising the equation completely

-0.05 (x² - 26x - 120) = 0
-0.05 (x+4) (x-30) = 0 
(x+4) (x-30) = 0

Question 8
 
Solving the equation

x+4 = 0 and x-30=0
x=-4 and x=30 

Question 9 

The roots of the equation is x = -4 and x = 30

Question 10

Please refer to the third graph

Question 11

Yes

Question 12

The negative zero means the initial distance of the cannon, where it was fired from

Question 13

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 30 units from the firing point, the cannon will fly pass it. 

Answer 2

The motion of the projectile (cannonball), given by the (quadratic)

polynomial function is the path of a parabola.

Response:

1. Vertical height
2. The point the projectile is at ground level
3. Yes, because is it where the projectile will land
4. x² - 26·x - 120 = 0
5. (x - 30)·(x + 4) = x² - 26·x - 120
6. p = -30, and q = 4
7. -0.05·(x² - 26·x - 120) = -0.05·(x - 30)·(x + 4)
8. x - 30 = 0 and x + 4 = 0 Which gives; x = 30 or x = -4
9. x = 30 or x = -4
10. The zeros of the function are x = 30, and x = -4
11. Yes
12. The negative zero (x = -4) means that the ground level is a point before the location at which the cannonball is launched (x = 0). It also means that the cannonball was launched above ground level.
13. Yes, because the projectile hits the ground 30 feet from the cannon

Which methods can be used to evaluate the polynomial?

1. f(x) is the output of the projectile function, where;

x = The input which is the distance

Therefore;

• f(x) = The vertical distance (height) of the cannonball at point x.

2. 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. The net should be placed where the cannonball is expected to reach ground level.

Therefore;

• The zeros indicates where the net should be placed

4. The given polynomial can be equated to 0 as follows;

x² - 26·x - 120 = 0

5. x² - 26·x - 120 = x² - 30·x + 4·x- 120

Which gives;

x·(x - 30) + 4·(x- 30) = (x - 30)·(x + 4)

• (x - 30)·(x + 4) = x² - 26·x - 120

6. Where; (x + p)·(x + q) = a·x² + b·x + c

We have;

The values of p and q that give the correct factors are;

• p = -30, and q = 4

7. The polynomial, -0.05·(x² - 26·x - 120) = 0 in factored form is therefore;

• -0.05·(x² - 26·x - 120) = -0.05·(x - 30)·(x + 4)

8. From the factors, we have;

-0.05·(x² - 26·x - 120) = -0.05·(x - 30)·(x + 4)

The roots of the equations, are given as follows;

x - 30 = 0 and x + 4 = 0

Which gives;

• x = 30 or x = -4

9. The roots of the equation are therefore;

• x = 30, and x = -4

10. The zeros for the function, are the values of x at which f(x) = 0

At x = 30 f(30) = -0.05 × (30² - 26 × 30 - 120) = 0

At x = -4, f(-4) =  -0.05 × ((-4)² - 26 × (-4) - 120) = 0

• The zeros of the function are x = 30, and x = -4

11. Yes. The zeros are the same as the roots found given that at the roots, f(x) = 0

12. 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.

What the negative zero means are therefore;

• It is a location before the point where the cannonball is launched
• The cannonball is launched from a height above ground level

13. At 30 feet, x = 30, which is a zero of the function, and the cannonball

is at ground level, such that a net 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

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