Question

How do you graph the function `y=-1/4x^2+3` and identify the domain and range?

Answer 1

See below.

Explanation:

The key points for graphing will be the y axis intercept this is the value of the function when x = 0.

From `y = -1/4x^2 + 3`

When x = 0, y = 3.

Since there is no `x` term the axis of symmetry is the y axis.

Since the coefficient of `x^2` is negative the parabola will look like this `nnn`

Next find roots of `-1/4x^2 + 3 = 0`

`-x^2 + 12 = 0`

`x^2 = 12 => x = +-sqrt(12) = +-2sqrt(3)`

Roots are `( - 2sqrt(3) , 0)` and `( 2sqrt(3) , 0 )`

The function is continuous so there are no restriction on `x` therefore the domain is:

`{x in RR}`

The maximum value occurs when `x = 0`. because the coefficient of `x^2` is negative. This is 3

as `x-> +-oo`

`-1/4x^2 + 3 -> -oo`

So range is

`{y in RR|-oo < y <= 3}`

See graph:

graph{-1/4x^2 + 3 [-10, 10, -5, 5]}