Problem 23 - Algebra 2

Question

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=-x^{2}-9 $$

Step-by-Step Solution

Answer

The function has a maximum value of -9 at its vertex, domain is \((-\infty, \infty)\), and range is \((-\infty, -9]\).

Step 1: Identify the Type of Function

The given function is \( f(x) = -x^2 - 9 \), which is a quadratic function of the form \( ax^2 + bx + c \). Here, \( a = -1 \), \( b = 0 \), and \( c = -9 \). The parabola opens downwards because \( a < 0 \). Hence, the function has a maximum value.

Step 2: Find the Vertex

The vertex form of a quadratic function \( ax^2 + bx + c \) helps in determining the maximum or minimum. The x-coordinate of the vertex is \( x = -\frac{b}{2a} \). Substituting \( b = 0 \) and \( a = -1 \) gives \( x = 0 \). Substitute \( x = 0 \) back into the function to get the maximum value: \( f(0) = -0^2 - 9 = -9 \).

Step 3: Determine the Domain

For any quadratic function, the domain is all real numbers because there are no restrictions on the values \( x \) can take. Therefore, the domain is \( (-\infty, \infty) \).

Step 4: Determine the Range

Since the parabola opens downwards and the vertex represents the maximum point, the maximum value of the function is \( f(x) = -9 \). Thus, the range is all real numbers \( y \) such that \( y \leq -9 \). This means the range is \( (-\infty, -9] \).

Key Concepts

Maximum and Minimum Values

In a quadratic function like \( f(x) = -x^2 - 9 \), understanding whether it has a maximum or minimum value is crucial. This is determined by the coefficient \( a \) in the quadratic equation \( ax^2 + bx + c \). In our function, \( a = -1 \). A negative \( a \) indicates that the parabola opens downward.

This means the vertex of the parabola is at its highest point, providing a maximum value. In contrast, if \( a > 0 \), the graph opens upwards, and the vertex would represent a minimum value. Hence, for \( f(x) = -x^2 - 9 \), the maximum value of the function is \(-9\), which is verified by finding the vertex.

Domain and Range

Understanding the domain and range of a function helps in grasping the behavior of quadratic equations. The domain of quadratic functions like \( f(x) = -x^2 - 9 \) is all real numbers. This is expressed as \( (-\infty, \infty) \). It means that we can plug in any real number for \( x \) without restriction.

Next, the range is a bit different. Since our function \( f(x) = -x^2 - 9 \) has a maximum point at \(-9\), the range includes all real numbers less than or equal to this maximum value. Thus, the range can be expressed as \( (-\infty, -9] \), highlighting that the function's output does not go above -9.

Vertex Form

The vertex form of a quadratic function is a powerful way to identify the vertex, which helps determine the maximum or minimum values and analyze the graph's direction. The general form is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

In the standard form \( ax^2 + bx + c \), the x-coordinate of the vertex can be calculated by \( x = -\frac{b}{2a} \). Applying this formula to \( f(x) = -x^2 - 9 \) with \( a = -1 \) and \( b = 0 \), we find the vertex at \( (0, -9) \).

Thus, this vertex form allows us to confidently determine that the maximum value is \(-9\), and the vertex of the parabola is precisely at \( x = 0 \). Knowing how to find the vertex is key in interpreting the function's graph effectively.