You measure of the distance traveled that is increasing its speed at a constant rate. What is the quadratic function that models distances of 21 ft at 1 s, 59 ft at 2 s, and 141 ft at 4 s?

Answer 1

The general formula for a quadratic function is y = ax² + bx + c

We also have points on this function: (1, 21), (2, 59), and (4, 141)

Plug these points into the function

21 = a(1)² + b(1) + c

59 = a(2)² + b(2) + c

141 = a(4)² + b(4) + c

Simplify the equations:

21 = a + b + c

59 = 4a + 2b + c

141 = 16a + 4b + c

Now, we'll solve for a, b, and c.

We'll do equation 2 - equation 1:

38 = 3a + b

Now we'll do equation 3 - equation 2:

82 = 12a + 2b

We can divide both sides of the equation by 2

41 = 6a + b

We'll rewrite the two new simplified equations and solve for a and b:

38 = 3a + b

41 = 6a + b

Subtract the equations:

-3 = -3a. a = 1

Plug 1 back in for a and solve for b:

38 = 3(1) + b

b = 35.

Lastly solve for c: (We'll use the very first equation)

21 = a + b + c

21 = (1) + (35) + c

21 = 36 + c

c = -15

So a = 1, b = 35, and c = -15

Therefore, the equation of the quadratic is y = 1x² + 35x - 15