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

Question

Question #1:

The graph shows a function f that is a parabola.

a) Determine f(x).

b) Determine the domain and range of f^-1(x).

c) Determine the equation(s) for .

Question #2:

A highway tunnel has a shape that can be modelled by the equation of a parabola. The tunnel is 18 m wide and the height of the tunnel 16 m from the edge is 5 m.

a) Determine the equation of the parabola.

b) Sketch a graph of your parabola. Capture your work and insert the image or link here. :)

c) Can a truck that is 8 m tall and 4 m wide pass through the tunnel? Justify your decision.

Answer 1

y=a(x+4)^2 +3 is the parabola's equation in vertex form with vertex = maximum point = (-4,3)

to solve for a, the coefficient of the x^2 term, we need one more point on the parabola, plug it in and simplify

use (-2,-1), x=-2, y=-1

-1 = a(-2+4)^2 +3 = 4a +3

a = -4/4 = -1

y = f(x) = -(x+4)^2 +3 is the parabola's equation

to find f^-1(x) switch x and y and solve for the new y

x =-(y+4)^2 + 3

solve for y

3-x = (y+4)^2

take the square root of both sides

y+4 =+ or - sqr(3-x)

y =+ or - sqr(3-x) - 4 is the equation of the inverse relation

f^-1(x) = + sqr(3-x) -4

range is any real value, (-infinity, + infinity)

domain is x<3 or the interval (3, infnity)

It's a rightward opening parabola with vertex (3, -4)