Problem 40 - Algebra and Trigonometry

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. $$ f(x)=2 x^{2}-8 x-3 $$

Step-by-Step Solution

Answer

The function has a minimum value of -7 at x-value of 2. The domain is all real numbers and the range is \([-7, \infty)\).

Step 1: Determine whether the function has a minimum or maximum value

The coefficient of \(x^{2}\) in the given function, \(f(x)=2x^{2}-8x-3\), is 'a' = 2, which is positive. This implies the function opens upward, so it has a minimum value, not a maximum.

Step 2: Find the minimum value and where it occurs

To find the minimum value, we need to find the vertex of the parabola. The x-coordinate of the vertex is given by \(-b/{2a}\). Substituting \(b=-8\) and \(a=2\) we get \(x=-(-8)/2*2 = 2\). Substituting \(x=2\) in the function \(f(x)\), we get \(f(2) = 2*2^{2} - 8*2 -3 = -7\). So the minimum value is -7, and it occurs at \(x=2\)

Step 3: Identify the function's domain and range

The domain of any quadratic function is the set of all real numbers. So the domain of \(f(x) = 2x^{2} - 8x -3\) is all real numbers. The function opens upwards and has a minimum value of -7. Thus, the range of the function is \([-7, \infty)\)

Key Concepts

Minimum Value in Quadratic Functions

In a quadratic function, the minimum value is a crucial characteristic. It signifies the lowest point on the graph of the function when it opens upwards. This particular quadratic function, represented by \(f(x) = 2x^2 - 8x - 3\), displays an upward opening because the coefficient of \(x^2\), which is 2, is positive.
Whenever the coefficient of \(x^2\) is positive, the parabola has a minimum point. This knowledge can quickly help you determine whether the function will have a minimum value without graphing the function. In contrast, if the coefficient were negative, the parabola would open downwards, resulting in a maximum point.
Understanding the minimum value is key to analyzing quadratic functions, as it tells us the lowest output value that the function can reach. This provides useful insights into the behavior of the function, crucial for real-world applications where you wish to minimize certain quantities, such as cost or time.

Vertex of a Parabola

The vertex of a parabola is the point where the minimum or maximum value of a quadratic function occurs. For a given quadratic function \(f(x) = ax^2 + bx + c\), the vertex can be found at the x-coordinate given by \(-\frac{b}{2a}\).
Substituting the values from our function, where \(a = 2\) and \(b = -8\), we calculate

• \(x = -\frac{-8}{2 \times 2} = 2\)

This x-value tells us the horizontal location where the vertex sits. To find the corresponding y-coordinate of the vertex (which is the minimum value in this function), substitute \(x = 2\) back into the function:

• \(f(2) = 2(2)^2 - 8(2) - 3 = -7\)

This means the vertex of the parabola, and thus the minimum point, is at \((2, -7)\).
The vertex provides valuable information about the function, indicating its turning point and giving insight into other properties like axis of symmetry, which is a vertical line passing through the vertex.

Domain and Range of Quadratic Functions

The domain and range describe the set of inputs and outputs a function can take. For quadratic functions like \(f(x) = 2x^2 - 8x - 3\), the domain is very straightforward.
The domain of any quadratic function is the set of all real numbers. This is because a quadratic function is defined for every real number x. In terms of the graph, it means that you can plug any x-value into the function to get a corresponding y-value.

The range, however, depends on whether the parabola has a maximum or minimum value. Since this function has a minimum value at -7, the parabola opens upwards. Thus, the range is always above or equal to this minimum point. Mathematically, we write this as the interval \([-7, \infty)\).

• Domain: All real numbers (\(-\infty, \infty\))
• Range: \([-7, \infty)\)

Knowing the domain and range helps you understand what values a function can output, providing a complete picture of its behavior, and is essential when solving real-world problems using quadratic equations.