Question

Based on the graph of b(x)=−x2−8x−9
, write the range in inequality form.(1 point)
The range in inequality form is y≤
.

Answer 1

The function \( b(x) = -x^2 - 8x - 9 \) is a quadratic function that opens downwards (since the coefficient of \( x^2 \) is negative). To determine the range, we first need to find the vertex of the parabola, as the vertex will give us the maximum value of the function.

We can find the vertex using the formula for the x-coordinate of the vertex of a quadratic function \( ax^2 + bx + c \), which is given by:

\[
x = -\frac{b}{2a}
\]

Here, \( a = -1 \) and \( b = -8 \).

\[
x = -\frac{-8}{2 \times -1} = \frac{8}{-2} = -4
\]

Now, plug this value back into the function to find the y-coordinate of the vertex:

\[
b(-4) = -(-4)^2 - 8(-4) - 9
\]
\[
b(-4) = -16 + 32 - 9
\]
\[
b(-4) = 16 - 9 = 7
\]

The vertex of the parabola is at the point \((-4, 7)\). Since the parabola opens downwards, the maximum value of \( b(x) \) is 7.

Thus, the range of \( b(x) \) in inequality form is:

\[
y \leq 7
\]