The graph shows a function f that is a parabola. Determine the following...

Question

Question #1:

The graph shows a function f that is a parabola.

a) Determine the equation of f(x).

b) Determine the equation(s) of .

Question #2:

Muhammad is disc shooting. The height of the disc, in metres, after t seconds, is modelled by the function  . The path of Muhammad's pellet is modelled by the function, with the same units.

a) How high off the ground will the disc be when it is hit?

b) After how many seconds will the disc be hit?

Answer 1

in vertex form the equation of the parabola is: y = a(x-1)^2 + 2

the y intercept = 4, where the parabola intersects the y axis

y = a(x^2 -2x +1) + 2

y = ax^2 -2ax + a + 2

a +2 = 4

a = 4-2 = 2

a = 2

the equation is

y =f(x) = 2(x-1)^2 + 2

the inverse "function" is not the inverse relation, but half of it, either the bottom or top half, of a horizontal parabola, 2 possibilities

switch x and y, then solve for the new y

x = 2(y-1)^2 + 2

(x-2)/2 = (y-1)^2

take the square root of both sides

y-1 = + or - square root of [(x-2)/2)]

y = sqr[(x-2)/2] +1 or -sqr[(x-2)/2] +1

y= sqr[(x-2)/2} +1 is the top half of a right ward opening parabola with vertex (2, 1)

y = -sqr[(x-2)/2] + 1 is the bottom of a leftward opening parabola with the same vertex (2,1)

either equation is an inverse function, but of only half the original parabola's function.

IF you try to use both, with plus and minus the square root, it's no longer a fuction, as it fails to pass the vertical line test.

the "f" suggests a function and inverse function

but if you interpret is as merely an inverse relation, then f^-1(x) = + sqr((x-2)/2) + 1

h(t) =-4.9t^2 + 32t + 2

g(t) = 28.5t + 1

set them equal and solve for t

-4.9t^2 +32t +2 = 28.5t + 1

-4.9t^2 + 3.5t +1 = 0

use the quadratic formula

t = 3.5/9.8 + (1/9.8)sqr(3.5^2 + 4(4.9))

t = 3.5/9.8 + (1/9.8)sqr(12.25 + 19.6)

t = 3.5/9.8 + (1/9,8)sqr(31.85)

t= 3.5/9.8 + 5.6435/9.8

t = 9.144/9.8

t = .933 seconds

t = almost 1 second

h(.933) = -4.9(,933)^2 + 32(.933) +2 = -4.265 + 29.856 +2 = -4.265 +29.856+2 = 31.856- 4.265 = 27.591 meters high

g(.933) = 28.5(.933) +1 = 26.593+1 = 27.591

about 27.591 meters high when the pellet hits the disc