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)
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)
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Detailed analysis:
The scenario mentions that a mouse covered a total of 3/24 miles over a span of 3 hours. The mouse traveled an identical distance during each hour. To find out how far the mouse went in one hour, we need to divide 3/24 miles by 3. Thus,
Matt's calculation indicates that the outcome is 3/8 hours. When he divides 3/24 by 3, he should take the reciprocal of 3, but he doesn't, resulting in 9 as the numerator and 24 as the denominator. The fraction 9/24 simplifies to 3/8. However, the accurate answer should have been 1/24 hours.