Write a quadratic equation for the following scenarios a A b

Write a quadratic equation for the following scenarios:

a) A boll is tossed in the air from height of 5 feet and the following data is recorded:

Time (seconds)

0

0.25

0.5

0.75

1.0

1.25

Height (feet)

5

8

9

8

5

0

b) The vertex is (-5,100) and the vertical intercept is 0

c) The x-intercepts are -2 and 7 and the y-intercept is 6

Solution

a) Lets take a standrad quadratice equation h = at^2 +bt +c

Now substitute the points given to find a, b, c

5 = a*0 +b*0 +c----> c=5

Now ( 0.25, 8) 8 = 0.25^2a +0.25b +5 ----(1)

3 = 0.0625a +0.25b

(1.0, 5) 5 = a +b +5 -----(2)

0 = a+b

Solve equation 1 and 2 to get values of a and b:

a = -16 ; b =16

So, quadratic equation : h(t) = -16t^2 +16t +5

a)   Vertex form: y-k = a(x-h)^2

where (h,k) is vertex,So, (h, k) =(-5, 100) we also another point vertical intercept is 0 ---( 0, 0)

y -100 = a(x +5)^2

0 -100 = a(5^2) ----> a = -4

y -100 = -4(x +5)^2

c) Let standard quadratic equation be y = ax^2 +bx +c

x intercepts: ( -2. 0) and ( 7, 0)

y intercepts : ( 0,6)

use these points to solve for a, b, c:

0 = 4a -2b +c

0 = 49a +7b +c

6 = a*0 +b*0 +c ----> c = 6

So, solve these two equations:

4a -2b =-6

49a +7b =-6

On solving we get : a = -3/7 ; b= 15/7

So, y = -3x^2/7 +15x/7 +6