Problem 22 - College Algebra with Corequisite Support

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: all real numbers; Range: \( y \leq -6 \).

Step 1: Understand the Structure of the Function

We have a quadratic function given by \( f(x) = -2(x+3)^2 - 6 \). This is a vertex form of a quadratic equation, which shows a parabola opening downward due to the negative sign in front of the \((x+3)^2\) term.

Step 2: Determine the Domain

The domain of any quadratic function is all real numbers, since you can substitute any real number for \(x\) and perform the calculations without any restrictions.

Step 3: Find the Vertex

The function is in the form \( f(x) = a(x-h)^2 + k \), which indicates the vertex is at \( (h, k) \). Here, \( h = -3 \) and \( k = -6 \), so the vertex is at \(-3, -6\).

Step 4: Determine the Range

Since the parabola opens downward (because the coefficient of \((x+3)^2\) is negative), the range is all real numbers less than or equal to the y-coordinate of the vertex. Thus, the range is \( y \leq -6 \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (typically denoted as "x") that a function can accept. For quadratic functions, the domain is quite straightforward. You can substitute any real number into the quadratic equation, and it will yield a corresponding output.

In our case for the function, \( f(x) = -2(x+3)^2 - 6 \), the domain is all real numbers. This means you can safely substitute any real number for \( x \), such as 0, -5, 2.5, or even expressions like \( \sqrt{2} \) or \( \pi \).

To put it simply:

• The graph of a quadratic function stretches infinitely to the left and right on the x-axis.
• This results in there being no restrictions on \( x \), hence the domain is all real numbers, often expressed in interval notation as \((-fty, \infty)\).

Range of a Function

The range of a function consists of all possible output values (typically denoted as "y") that are produced by the function for all values in the domain. Unlike the domain, the range of a quadratic function is affected by its specific form and properties.

In vertex form, which is \( f(x) = a(x-h)^2 + k \), the position and direction of the parabola determine the range. In our function \( f(x) = -2(x+3)^2 - 6 \):

• The parabola opens downwards because the coefficient \( a = -2 \) is negative.
• The vertex of the parabola acts as the peak or maximum point because the opening is downwards.

Given the vertex is at \( (-3, -6) \), no value of \( y \) will exceed \(-6\). Therefore, the range is \( y \leq -6 \), which includes all real numbers less than or equal to \(-6\). This is expressed in interval notation as \((-fty, -6] \).

This understanding is essential as:

• It helps in visualizing the extent of the graph on the y-axis.
• It indicates that the y-values decrease as they stretch towards \(-\infty\), but never rise above the vertex.

Vertex Form

Vertex form is a particular way of expressing a quadratic function that emphasizes its geometric features, particularly the vertex. A quadratic function can be written in vertex form as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Understanding vertex form is beneficial as it reveals information about the graph intuitively:

• \( a \) determines the direction and the width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
• The vertex \((h, k)\) offers a clear point of graph symmetry and the highest or lowest point depending on the parabola's direction.

In the given function \( f(x) = -2(x+3)^2 - 6 \), you can identify:

• The vertex as \( (-3, -6) \), indicating the parabola undergoes a horizontal shift of 3 units left and a vertical shift of 6 units down from the origin.
• Since \( a = -2 \), this parabola opens downward and is more narrow compared to a standard parabola owing to the stretching factor.

Recognizing these traits makes it easier to graph quadratic functions and understand their properties visually and algebraically.