Problem 42 - College Algebra

Question

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)--2 x^{2}-12 x+3$$

Step-by-Step Solution

Answer

The quadratic function \(f(x)=-2x^2-12x+3\) has a maximum value, not a minimum. This maximum value is -3, and it occurs at \(x=3\). The domain of the function is all real numbers \(-\infty < x < \infty\), and the range is all real numbers lesser than or equal to -3, \(-\infty < y \leq -3\).

Step 1: Identifying the nature of the function

First, determine the nature of the quadratic function, whether it has a minimum value or a maximum value. A quadratic function is of the form \(f(x) = ax^2 + bx + c\). If \(a\) (the coefficient of \(x^2\)) is positive, the function has a minimum value, and if \(a\) is negative, the function has a maximum value. In this case, \(a = -2\), which is a negative value. Therefore, this function has a maximum value.

Step 2: Finding the maximum value and where it occurs

The maximum (or minimum) value of a quadratic function occurs at its vertex. The x-coordinate of the vertex can be found using the formula \(-b/2a\). Substituting \(a = -2\) and \(b = -12\) gives the vertex at \(x=3\). To find the maximum value of the function, substitute \(x=3\) into the function: \(f(3)= -2*(3)^2 - 12*3 + 3 = -3\). So, the function has a maximum value of -3 which occurs at \(x=3\).

Step 3: Identifying the function's domain and its range

The domain of a quadratic function is all real numbers since it is defined for any real number. Therefore, the domain is \(-\infty < x < \infty\). Quadratic functions have ranges that are all the values that the function can take, depending on whether the function has a maximum or minimum value. As the function has a maximum value (established in Step 1), the range is \(-\infty < y \leq -3\)

Key Concepts

Vertex of a Quadratic Function

In a quadratic function, the vertex is one of the most significant points. It is the point where the curve changes direction. For the function given by \[f(x) = ax^2 + bx + c\] the vertex can be identified using the formula for the x-coordinate, which is \[-\frac{b}{2a}\]. This would give you the position on the x-axis where the vertex is located. Once you find that x-coordinate, you can plug it back into the function to find the corresponding y-coordinate.

For instance, with the quadratic function \[-2x^2 - 12x + 3\], the vertex x-coordinate will be found by calculating \[-\frac{-12}{2 \times -2} = 3\]. By substituting \(x = 3\) into the function, you'll find the y-coordinate of the vertex, which for this function is -3. Hence, the vertex of this quadratic function is \((3, -3)\). The vertex tells us a lot, including whether the function has a maximum or minimum value.

Domain of a Function

The domain of a quadratic function is generally straightforward. It represents the set of all possible inputs (x-values) that your function can take. For most quadratic functions, the domain is all real numbers, which can be written as\(-\infty < x < \infty\). This means you can substitute any real number into the quadratic expression and you will get a valid output.

So for the quadratic function \(-2x^2 - 12x + 3\), there are no restrictions on the values that x can take, hence the domain remains all real numbers. This wide range of inputs is a characteristic trait of quadratic functions in general, barring any specific modifications or context-restraints imposed in a specific problem.

Range of a Function

Understanding the range of a quadratic function involves determining all the possible outputs (y-values) that the function can produce. The range is determined by the vertex and whether the parabola opens upwards or downwards.

For a quadratic function in general form, \[f(x) = ax^2 + bx + c\] if \(a > 0\), the parabola opens upwards and thus has a minimum y-value at the vertex. Conversely, if \(a < 0\), as in \(-2x^2 - 12x + 3\), the parabola opens downwards, providing a maximum y-value at the vertex.

In this exercise, since the parabola opens downward, the highest y-value is at the vertex, which is \(-3\). Therefore, the range of the function is written as\(-\infty < y \leq -3\). This notation shows that the function takes on all y-values below or equal to the vertex's y-value.

Maximum and Minimum Values

For quadratic functions, understanding when a function reaches a maximum or minimum is vital when analyzing the behavior of the graph.

With the quadratic formula \(f(x) = ax^2 + bx + c\), the coefficient \(a\) indicates whether our function has a maximum or a minimum:

• If \(a > 0\), the parabola opens upwards and the vertex will be at the minimum value of the function.
• If \(a < 0\), the parabola opens downwards, thereby making the vertex the maximum point.

In this problem, as seen with \(-2x^2 - 12x + 3\), the function opens downward because \(a = -2\) (a negative value). This means that the function has a maximum value at the vertex.

Calculating this provides the x-coordinate (\(-\frac{-12}{2 \times -2} = 3\)), and when substituted into the function, gives the y-coordinate (-3). Therefore, the maximum value is -3, occurring at x = 3. While quadratic functions typically have either a maximum or minimum, this is firmly dictated by the direction of their opening.