Problem 22 - College Algebra

Question

For the following exercises, determine the domain and range of the quadratic function. $$ f(x)=-2(x+3)^{2}-6 $$

Step-by-Step Solution

Answer

Domain is \((-\infty, \infty)\); Range is \((- fty, -6]\).

Step 1: Understand the Function Structure

The given function is a quadratic function in the form \( f(x) = a(x-h)^2 + k \), where \(a = -2\), \(h = -3\), and \(k = -6\). This is a parabola that opens downwards because \(a\) is negative.

Step 2: Determine the Domain

The domain of a quadratic function is all real numbers because there is no restriction on the values that \(x\) can take. Thus, the domain is \((-\infty, \infty)\).

Step 3: Find the Vertex of the Parabola

The vertex of the parabola is at the point \((h, k)\). Here, \(h = -3\) and \(k = -6\), so the vertex is \((-3, -6)\).

Step 4: Determine the Range

Since the parabola opens downwards (\(a = -2 < 0\)), the highest point on the graph is the vertex. Thus, the range of \(f(x)\) is all \(y\) values less than or equal to \(k\), which is \(-6\) in this case. Therefore, the range is \((-fty, -6]\).

Key Concepts

Domain of a Function

A quadratic function is symbolized by the expression \( f(x) = ax^2 + bx + c \) or transformed versions like \( f(x) = a(x-h)^2 + k \). Regardless of the shape or direction of a parabola, the domain is always about the possible \( x \)-values the function can accept.
In the case of any quadratic function including \( f(x)=-2(x+3)^{2}-6 \), the domain is rather straightforward. For quadratics, it extends across all real numbers because there are no restrictions that prevent \( x \) from taking on any value. With no holes, breaks, or asymptotes in a quadratic function graph, these functions cover the entire \( x \)-axis.

• This means the domain is \((-fty, fty)\).
• Every quadratic, regardless of how it is shifted or opened up or down, has the same domain.

Understanding the domain is essential as it lays the foundation for how the function behaves horizontally across the graph.

Range of a Function

While the domain of a quadratic function is all-encompassing, the range involves the set of possible \( y \)-values the function produces. This is heavily influenced by two main factors: the orientation of the parabola (up or down) and the vertex's \( y \)-value.
In a function like \( f(x)=-2(x+3)^{2}-6 \), we notice a few things. The coefficient \( a = -2 \) is negative, indicating that the parabola opens downward. Therefore, its topmost point is the vertex, and the parabola stretches downward without bound.

• The vertex being the highest point limits the range.
• For this function, at the vertex \((-3, -6)\), the \( y \)-value is \(-6\). This means the function can produce \( y \)-values as low as \(-fty\) but no higher than \(-6\).
• Thus, the range is \((-\infty, -6]\).

Knowing the range is crucial for understanding the vertical span and constraints the function places on its values.

Vertex of a Parabola

The vertex is a pivotal feature in understanding quadratics as it conveys critical information about the graph itself. It is the point \((h, k)\) in the form \( f(x) = a(x-h)^2 + k \), around which the parabola is symmetrically organized.
For our quadratic function \( f(x)=-2(x+3)^{2}-6 \):

• The vertex is located at \((-3, -6)\).
• This vertex indicates two things: the \( x \)-coordinate \( h = -3 \) is the axis of symmetry, dividing the parabola into two mirror-image halves, and \( k = -6 \) represents either the minimum or maximum \( y \)-value.
• Since the function opens downwards, \(-6\) is the maximum value.

In summary, the vertex is not just a point—it represents the peak or trough of the parabola and stabilizes how the function behaves overall.