What is the X Coordinate of the Turning Point for the Graph of the Function f(x) 3x^2 - 2x - 7?

The function given is f(x) 3x^2 - 2x - 7. To determine the x-coordinate of the turning point of this graph, we can use the method of differentiation or apply the formula for the vertex of a parabola. In this article, both methods will be explained in detail to help you understand the process.

Method 1: Using the Vertex Formula

The x-coordinate of the vertex (turning point) of a parabola in the form ax^2 bx c can be found using the formula:

$x -frac{b}{2a}$

For the function f(x) 3x^2 - 2x - 7, the coefficients are:

a 3 b -2 c -7

Now, apply the formula:

$x -frac{-2}{2 cdot 3} frac{2}{6} frac{1}{3}$

Therefore, the x-coordinate of the turning point is x frac{1}{3}.

Method 2: Using Differentiation

Another approach is to use calculus by finding the first derivative of the function and setting it equal to zero. The first derivative of f(x) 3x^2 - 2x - 7 is:

$f'(x) 6x - 2$

Set the first derivative equal to zero to find the turning point:

$6x - 2 0$

Solve for x by isolating it:

$6x 2 x frac{2}{6} frac{1}{3}$

This confirms that the x-coordinate of the turning point is x frac{1}{3}.

Understanding the Parabola

The function f(x) 3x^2 - 2x - 7 represents a parabola opening upwards (since the coefficient of x^2 is positive). The turning point, also known as the vertex, is the minimum point of the parabola. For a parabola that opens upwards, the vertex is the lowest point on the graph.

Significance of the Turning Point

The turning point of the parabola is significant because it represents the maximum or minimum value of the function. In this case, since the parabola opens upwards, the turning point is the minimum value of the function. Knowing the turning point is important in various applications such as optimization problems, physics, and engineering.

Conclusion

In summary, the x-coordinate of the turning point for the function f(x) 3x^2 - 2x - 7 is x frac{1}{3}. This can be determined using either the vertex formula or by finding the first derivative and setting it to zero. Understanding the turning point is crucial for analyzing the behavior of quadratic functions and their graphical representations.

Key Takeaways

The turning point, or vertex, of a parabola described by ax^2 bx c is given by x -frac{b}{2a}. The first derivative can also be used to find the x-coordinate of the turning point. The turning point of a parabola is significant as it represents the maximum or minimum value of the function.

Understanding these concepts is essential for advanced studies in calculus and is valuable in various scientific and engineering applications. If you have any further questions or need more detailed explanations, feel free to explore related topics or seek assistance from a math tutor.