Determining the Axis of Symmetry and Optimum Value of a Function: fx x^2 - 4x

In this article, we will explore how to determine the axis of symmetry and the optimal value of the quadratic function f(x) x^2 - 4x. We will also examine some general facts about parabolic functions and showcase the process of completing the square for this particular function.

Understanding the Function: f(x) x^2 - 4x

Let's start with the given function f(x) x^2 - 4x. This is a quadratic function, and it produces a parabolic curve when graphed. The standard form of a quadratic function is f(x) ax^2 bx c, where a, b, and c are constants. In this case, a 1, b -4, and c 0.

Determining the Axis of Symmetry

The axis of symmetry for a parabola can be calculated using the formula x -b / (2a). In the function f(x) x^2 - 4x, a 1 and b -4. Substituting these values into the formula gives:

x -(-4) / (2 * 1) 2

This means the axis of symmetry of the parabola is at x 2.

Completing the Square

Another method to determine the optimal value and the vertex (intersection of the vertex with the axis of symmetry) is by completing the square. This technique transforms the quadratic equation into the vertex form, which is y a(x - h)^2 k, where (h, k) is the vertex of the parabola.

Let's rewrite the function f(x) x^2 - 4x in vertex form:

First, take the coefficient of x (which is -4), divide it by 2, and then square it. This gives (-4/2)^2 4. Add and subtract this square inside the function:

x^2 - 4x 4 - 4 (x - 2)^2 - 4

Thus, the function can be expressed as f(x) (x - 2)^2 - 4.

From this vertex form, we can see that the vertex of the parabola is at (2, -4). The axis of symmetry is then x 2, and the minimum value of the function is -4, since the parabola opens upwards.

General Facts About Upright Parabolic Functions

Parabolic functions often have a vertical opening and an upward direction. This means the parabola opens upwards, and the vertex is the minimum point.

The completed square method found that the function f(x) x^2 - 4x - (1/2) can be expressed as f(x) (x - 2)^2 - 7/2. This confirms that the vertex of the parabola is at (2, -7/2) and the axis of symmetry is x 2.

From the vertex form, it can be clearly seen that the minimum value of the function f(x) x^2 - 4x - (1/2) is -7/2, as the parabola opens upwards and the vertex is the lowest point on the curve.

Conclusion

In summary, determining the axis of symmetry and the optimal value of the function f(x) x^2 - 4x involves a combination of algebraic techniques such as completing the square. The axis of symmetry is x 2, and the optimal value (minimum value) is -7/2. Understanding these concepts is essential for effectively working with quadratic functions and analyzing their behavior.