Introduction to Parabolas: X-Intercepts and Vertex Identification

Parabolas are a fundamental component of mathematics, especially in algebra and calculus. The vertex and x-intercepts of a parabola are important characteristics that can provide valuable insights. In this article, we will explore how to find the x-intercepts and the coordinates of the vertex for the parabola defined by the equation y -x^2 2x - 1.

Step 1: Finding the X-Intercepts

The x-intercepts of a parabola occur when y 0. To find these, we set the equation equal to zero:

0 -x^2 2x - 1

x^2 - 2x 1 0

This equation can be factored as follows:

(x - 1)^2 0

Therefore, the solution is:

x - 1 0 implies x 1

Thus, there is one x-intercept at (1, 0). This point is also the vertex of the parabola.

Step 2: Finding the Vertex

The vertex of a parabola given by the standard form y ax^2 bx c has an x-coordinate that can be found using the formula:

x -frac{b}{2a}

In our given equation y -x^2 2x - 1, we identify a -1 and b 2. Substituting these values into the formula, we get:

x -frac{2}{2 cdot -1} 1

To find the y-coordinate of the vertex, we substitute x 1 back into the original equation:

y -1^2 2 cdot 1 - 1 -1 2 - 1 0

Therefore, the coordinates of the vertex are (1, 0).

Summary

For the parabola defined by the equation y -x^2 2x - 1:

The x-intercept is (1, 0). The coordinates of the vertex are (1, 0).

Since the vertex and the x-intercept are the same point, the parabola has a single point that is both the vertex and an x-intercept.

Conclusion

Understanding how to find the x-intercepts and the vertex of a parabola is crucial for analyzing and graphing quadratic equations. In this case, we found that the x-intercept and vertex for the parabola defined by y -x^2 2x - 1 are (1, 0). This information can be used to sketch the parabola or to solve more complex problems involving quadratic equations.