You take measurements of the distance traveled by an object that is increasing its speed at a constant rate. The distance traveled as a function of time can be modeled by a quadratic function.
a. Write a quadratic function that models distances of 10 ft at 1 sec, 30 ft at 2 sec, and 100 ft at 4 sec.
b. Find the zeros of the function.
c. Describe what each zero represents for this real-world situation.
a. In order to write a quadratic function that models the distances, we can use the standard form of a quadratic equation:
y = ax^2 + bx + c
Given the following distances and times:
- 10 ft at 1 sec (x = 1)
- 30 ft at 2 sec (x = 2)
- 100 ft at 4 sec (x = 4)
We can form three equations using these data points:
10 = a(1)^2 + b(1) + c
30 = a(2)^2 + b(2) + c
100 = a(4)^2 + b(4) + c
Simplifying these equations, we get the following system of equations:
a + b + c = 10
4a + 2b + c = 30
16a + 4b + c = 100
Solving this system of equations will give us the values of a, b, and c, which we can substitute back into the equation y = ax^2 + bx + c to get the quadratic function that models the distances.
b. The zeros of a quadratic function are the values of x where the function evaluates to zero. To find the zeros of the quadratic function we obtained in part a, we need to solve the equation:
ax^2 + bx + c = 0
Solving this quadratic equation will give us the values of x where the object has traveled a distance of zero.
c. In this real-world situation, each zero represents a specific time when the object has traveled a distance of zero. This can be interpreted as the time when the object is at rest or has not yet started moving, or when it has come to rest after traveling a certain distance.