Question

Identify the following key attributes of the polynomial

f(x)= – 4x2 + 9

What is the range?

Group of answer choices
y <= 9
y >= 0
y >= 1
y >= 9

Answer 1

To analyze the polynomial \( f(x) = -4x^2 + 9 \), we start by identifying its key attributes.

1. **Type of Polynomial**: This is a quadratic polynomial, which generally takes the form \( ax^2 + bx + c \). Here, \( a = -4 \), \( b = 0 \), and \( c = 9 \).

2. **Direction of Opening**: Since the leading coefficient \( a \) is negative (\(-4 < 0\)), the parabola opens downward.

3. **Vertex**: The vertex form for a quadratic function \( ax^2 + bx + c \) can be calculated using the vertex formula \( x = -\frac{b}{2a} \). For our polynomial:
- \( b = 0 \)
- \( a = -4 \)

Thus, the x-coordinate of the vertex is:
\[
x = -\frac{0}{2(-4)} = 0
\]

Now substitute \( x = 0 \) back into the function to find the y-coordinate of the vertex:
\[
f(0) = -4(0)^2 + 9 = 9
\]

Therefore, the vertex is \( (0, 9) \).

4. **Range**: Since the parabola opens downward and the vertex is the maximum point, the range of the function will encompass all values less than or equal to the y-coordinate of the vertex.

Thus, the range of the polynomial \( f(x) = -4x^2 + 9 \) is:
\[
y \leq 9
\]

**Answer**: y <= 9