Exploring the Maximum and Minimum Values of the Function f(x) x2 - x - 1

Understanding the behavior of functions is a fundamental aspect of calculus and algebra. In this article, we will explore the maximum and minimum values of the function f(x) x2 - x - 1. We will delve into the properties of this quadratic function, identify its vertex, and determine both the minimum value and the absence of a maximum value.

Introduction to Quadratic Functions

A quadratic function is a polynomial function of the form f(x) ax2 bx c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, which can open upwards or downwards. The orientation of the parabola (whether it opens upwards or downwards) depends on the sign of the coefficient a. If a > 0, the parabola opens upwards (concave up), and if a

Identifying the Maximum and Minimum Values

For the function f(x) x2 - x - 1, we first need to determine the orientation of the parabola. In this case, a 1, which is positive. Therefore, the parabola opens upwards (concave up). This means that the function has a minimum value but no maximum value.

The Minimum Value

The minimum value of a quadratic function occurs at the vertex of the parabola. The x-coordinate of the vertex of a parabola given by f(x) ax2 bx c is found using the formula x -b/(2a).

For the function f(x) x2 - x - 1, we have a 1 and b -1. Plugging these values into the formula gives:

x -(-1) / (2 * 1) 1 / 2 0.5

To find the minimum value of the function, we substitute x 0.5 back into the original equation:

f(0.5) (0.5)2 - 0.5 - 1 0.25 - 0.5 - 1 -1.25

Therefore, the minimum value of f(x) x2 - x - 1 is -1.25.

Graphing and Visualization

Graphing the function f(x) x2 - x - 1 can provide a visual representation of the parabola and its vertex. We plot the function on the x-y plane, with the x-axis representing the input values and the y-axis representing the output values.

Here is a plot of the graph:

The graph clearly shows the minimum point at (0.5, -1.25) and the parabola opening upwards.

Conclusion

In summary, for the quadratic function f(x) x2 - x - 1, the minimum value is -1.25, occurring at x 0.5. The function does not have a maximum value since the parabola opens upwards. Understanding the properties of quadratic functions is crucial for various applications in mathematics, physics, and engineering.

Related Keywords

Quadratic Function Minimum Value Maximum Value

Further Reading

For more detailed information on quadratic functions and their properties, we recommend the following resources:

Introduction to Quadratic Functions Vertex Form of Quadratic Functions Applications of Quadratic Functions

By delving further into these topics, you can gain a deeper understanding of quadratic functions and their applications in various fields.