Question

Question #02643

Answer 1

`f(x) <= 11`

Explanation:

Remember, the range of the function is the output of the function (i.e. The possible values for `f(x)`).

So basically, you want to find the vertex of the function as that is the maximum/minimum value of a quadratic function.

To find the vertex, you can complete the square. Here is my working for completing the square:

`-4x^2-8x+7`

`= -4(x^2 + 2x - 7/4)`

`= -4(x^2 +2x + (2/2)^2 - 7/4 - (2/2)^2)`

`= -4((x + 1)^2 - 7/4 - 1)`

`= -4((x + 1)^2 - 11/4)`

`= -4(x + 1)^2 + 11`

Therefore, using the vertex form (`a(x - p) + q`) the vertex is `(-1, 11)`

Using the `y`-coordinates, you can find the maximum value (as the `a` constant in the quadratic equation is a negative value).

Thus, the range is `f(x) <= 11`.

Hope that makes sense!