Question

What is the range for f(x) = 9 - 8x - x^2 .

Answer 1

The function is f (x ) = 9 - 8x - x ^2.

Let,  f (x ) = y = - x ^2 - 8x + 9.

The function y = - x ^2 - 8x + 9.

We first put the equation in to the form for a translated parabola y = a (x - h )^2 + k .

Center (h, k ).

Here x² coefficient is - 1, for perfect square make x² coefficient 1 by multiplying each side by - 1.

y = - x ^2 - 8x + 9

- y = x ^2 + 8x - 9

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x  coefficient = 8 then (half the x coefficient)² is (8/2)^2 = 16.

So, Add 16 to each side.

- y + 16 = x ^2 + 8x + 16 - 9

- y + 16 = (x + 4)^2 - 9

- y = (x + 4)^2 - 9 - 16 = (x + 4)^2 - 25

y = - (x + 4)^2 + 25.

The above function represents a parabola vertex form  y = a (x - h )^2 + k .

a = - 1, h = - 4, and k = 25.

a  is negative number the parabola opens down and has maximum value.

When the parabola opens down, it has a maximum point which is the vertex of parabola (- 4, 25).

In the maximum point y  = 25. so the graph of parabola cannot be greater than 25.

Thus, the range of function is y  ≤ 25.

Range of the function is  {y | y  ≤ 25}.