Problem 44 - 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. $$f(x)-6 x^{2}-6 x$$

Step-by-Step Solution

Answer

The given function has a maximum value. The maximum value is 1.5 and it occurs at x=-1/2. The domain of the function is (-∞, ∞), and the range is (-∞, 1.5).

Step 1: Type of function and extremum

The given function is quadratic (\(f(x)=-6x^{2}-6x\)), and since the coefficient of \(x^{2}\) is negative, this function has a maximum value.

Step 2: Maximum Value and x-value at which it occurs

To find the maximum value of the function and the x-value at which it occurs, we find the derivative of the function. The derivative of \(f(x)\) is given by \(f'(x)= -12x-6\). The maximum value occurs when the derivative is equal to zero. To find this x-value, we set \(f'(x)=0\) to get \(x = -1/2\). To find the maximum value of the function, we substitute this x-value back into the original function, \(f(-1/2)= -6*(-1/2)^{2}-6*(-1/2) = 1.5\)

Step 3: Function's Domain and Range

The domain of a quadratic function is the set of all real numbers, which can be represented as (-∞, ∞). The range of the function depends on whether the quadratic function has a maximum or minimum value. Since our function has a maximum value, the range will be (-∞, maximum value). Therefore, the range is (-∞, 1.5).

Key Concepts

Finding Maximum Value of Quadratic

To 'find the maximum value of a quadratic function', one must understand the shape and characteristics of a quadratic graph. A quadratic function typically takes the form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real numbers and \( a \eq 0 \). If \( a < 0\), the parabola opens downward, which means it has a maximum point at its vertex. The vertex can be found using the formula \( x = -\frac{b}{2a} \). Once you've found the x-coordinate of the vertex, plug it into the original function to get the maximum value of the function.
For the given function \( f(x) = -6x^2 - 6x \), since the coefficient of \( x^2 \) is \( -6 \), we can instantly tell that the parabola opens downward, indicating that our function has a maximum point. By applying the vertex formula, we get \( x = -\frac{-6}{2(-6)} = -\frac{1}{2} \). Plugging this x-value back into the equation, \(f(-\frac{1}{2}) = 1.5\), leads us to conclude that the maximum value of the function is 1.5, and it occurs when \( x = -\frac{1}{2} \).
By understanding this method, students can quickly find the maximum of any downward-opening quadratic function, which is a crucial concept in the analysis of quadratic equations.

Quadratic Function Domain and Range

When it comes to the 'quadratic function domain and range', all quadratic functions share a common trait: their domain is always all real numbers. Technically speaking, you can plug any real number into a quadratic function and receive an output. This domain is represented in interval notation as \( (-\infty, \infty) \).
However, the range varies depending on whether the quadratic has a maximum or minimum value. For functions with a maximum, such as our example \( f(x) = -6x^2 - 6x \), the range is limited from below by this maximum value. The range is all real numbers that are less than or equal to this maximum. Hence, for our function, since the maximum value is 1.5, the range is \( (-\infty, 1.5] \).
Understanding the domain and range is crucial for sketching graphs accurately and analyzing the behavior of functions. Always remember that the domain of quadratic functions is unrestricted, while the range is bounded by the vertex, either from above for functions that have a minimum value or from below for those with a maximum.

Derivative of Quadratic Function

The 'derivative of a quadratic function' plays a vital role in finding the function's extrema—its highs and lows. For a quadratic function \( f(x) = ax^2 + bx + c \), its derivative is given by the formula \( f'(x) = 2ax + b \).
By differentiating the function, one can solve for when \( f'(x) = 0 \) to find the x-values at which the function has its maximum or minimum values. This corresponds to the vertex of the parabola, which indicates the exact point where the slope of the tangent line is zero—meaning the graph levels out at its peak or trough.
For the given example \( f(x) = -6x^2 - 6x \), the derivative would be \( f'(x) = -12x - 6 \). Setting this equal to zero, \( -12x - 6 = 0 \) we find \( x = -\frac{1}{2} \). This confirms the x-value where the maximum occurs, as found by our earlier method. Mastering the concept of derivatives not only helps in analyzing quadratic functions but also in understanding broader aspects of calculus.