Problem 49 - 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. $$ f(x)=-\frac{1}{2} x^{2}-2 x+3 $$

Step-by-Step Solution

Answer

The function has a maximum value of -3; domain is \((-\infty, \infty)\), range is \((-\infty, -3]\).

Step 1: Recognize the Type of Function

The function is a quadratic equation of the form \( f(x) = ax^2 + bx + c \), where \( a = -\frac{1}{2} \), \( b = -2 \), and \( c = 3 \). Because \( a < 0 \), the parabola opens downwards, indicating that the function has a maximum value.

Step 2: Find the Maximum or Minimum Value

To find the vertex, which gives the maximum or minimum value of the quadratic function, use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = -\frac{1}{2} \) and \( b = -2 \). Substitute these values into the formula: \[ x = -\frac{-2}{2(-\frac{1}{2})} = \frac{-2}{-1} = 2. \] Now, substitute \( x = 2 \) back into the function to find \( f(2) \): \[ f(2) = -\frac{1}{2}(2)^2 - 2(2) + 3 = -\frac{1}{2}(4) - 4 + 3 = -2 - 4 + 3 = -3. \] Thus, the maximum value is \( -3 \) at \( x = 2 \).

Step 3: Determine the Domain of the Function

The domain of any quadratic function is all real numbers because there are no restrictions on the values that \( x \) can take. Therefore, the domain of \( f(x) = -\frac{1}{2} x^2 - 2x + 3 \) is \( (-\infty, \infty) \).

Step 4: Determine the Range of the Function

Since the function opens downwards and has a maximum value of \( -3 \), the range includes all values of \( y \) that are less than or equal to \( -3 \). Thus, the range is \( (-\infty, -3] \).

Key Concepts

Vertex Formula

Understanding the vertex formula is essential for analyzing quadratic functions. The vertex of a parabola, which represents the highest or lowest point, plays a crucial role in determining the function's maximum or minimum value. For any quadratic function in the standard form \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). This formula helps find the \( x \)-coordinate of the vertex. Once you have this \( x \)-value, you can determine the corresponding \( y \)-value by substituting the \( x \)-coordinate back into the original function. In the given function \( f(x) = -\frac{1}{2}x^2 - 2x + 3 \), the coefficients are \( a = -\frac{1}{2} \) and \( b = -2 \). Plug these into the vertex formula to find \( x = 2 \). Then by substituting \( x = 2 \) into the function, you get \( y = -3 \). Thus, the vertex is at \( (2, -3) \), indicating the parabola's maximum value because the parabola opens downward. This formula is a powerful tool for both graphing and analyzing quadratic functions.

Parabola Opening Direction

When analyzing quadratic functions, the opening direction of the parabola determines whether it has a maximum or minimum value. This is indicated by the sign of the coefficient \( a \) in the quadratic equation \( f(x) = ax^2 + bx + c \).

• If \( a > 0 \), the parabola opens upwards, forming a U-shape and possessing a minimum value at the vertex.
• If \( a < 0 \), as in our example function where \( a = -\frac{1}{2} \), the parabola opens downwards, forming an inverted U-shape and having its maximum value at the vertex.

This quick check tells you about the nature of the parabola, helping determine the kind of extremum (highest or lowest point) it possesses. Understanding the opening direction helps in visualizing the behavior of quadratic functions and predicting their graph shape upon analysis.

Domain and Range of Quadratic Function

Quadratic functions have straightforward and predictable domain and range values. The set of all possible \( x \) values a function can use is its domain, while the range is all possible \( y \) values the function can output. - **Domain:**For every quadratic function, the domain is always all real numbers \( (-\infty, \infty) \), as there are no restrictions for \( x \)-values.- **Range:**The range depends on whether the parabola opens upwards or downwards. Given that our example function \( f(x) = -\frac{1}{2}x^2 - 2x + 3 \) opens downward, the maximum value at the vertex is part of the range. Hence, the range includes all \( y \)-values that are less than or equal to the maximum value, forming the set \( (-\infty, -3] \). Understanding domain and range is fundamental in limiting or predicting outcomes of quadratic functions.