Understanding the X-Intercepts and Vertex of the Parabola (y x^2 - 8x - 12)

When analyzing the properties of a parabola, finding its x-intercepts and vertex is crucial for understanding its behavior. In this article, we will explore how to identify the x-intercepts and the coordinates of the vertex for the parabola given by the equation:

y x^2 - 8x - 12

1. Finding the X-Intercepts

The x-intercepts occur where the y-value is 0. This means we set the equation to 0 and solve for x:

0 x^2 - 8x - 12

Next, we factor the quadratic equation:

0 (x - 6)(x 2)

Setting each factor to 0 gives us:

x - 6 0 rarr; x 6

x 2 0 rarr; x -2

Thus, the x-intercepts are at the points (-2, 0) and (6, 0).

2. Finding the Vertex

The vertex of a parabola given by the equation y ax^2 bx c can be found using the formula for the x-coordinate of the vertex:

x -frac{b}{2a}

In this case, a 1 and b -8:

x -frac{-8}{2 cdot 1} 4

Now we substitute x 4 back into the original equation to find the y-coordinate of the vertex:

y 4^2 - 8 cdot 4 - 12 16 - 32 - 12 -20

Thus, the coordinates of the vertex are (4, -20).

Summary

X-intercepts: (-2, 0) and (6, 0) Vertex: (4, -20)

Additional Insights

The process of finding the vertex through completing the square can be seen as a reverse of the expansion:

y x^2 - 8x - 12 (x - 4)^2 - 20

Alternatively, Taking the derivative of the function f(x) x^2 - 8x - 12 and setting it equal to 0 gives us:

f'(x) 2x - 8 0

Solving for x:

2x 8 rarr; x 4

Substitute x 4 back into the function:

f(4) 4^2 - 8 cdot 4 - 12 16 - 32 - 12 -20

The vertex is at (4, -20).

Additionally, understanding the distance from the vertex to the focus and the directrix helps in further analyzing the parabola. For a parabola y x^2 - 8x - 12 with vertex (-4, -16) and a focus at (-4, -16 - c), where c is the distance, we can calculate:

4c^2 2^2 - c^2 rarr; 4c^2 4 - c^2 rarr; 5c^2 4 rarr; c 0.8

The focus is at (-4, -16 - 0.8) (-4, -16.8) and the directrix is y -16 0.8 -15.2. The distance from a point P(-2, 0) on the curve to the focus (-4, -16.8) is:

p^2 2^2 (0 16.8)^2 4 282.24 286.24 rarr; p 16.92

The distance from P to the directrix is 16.8, confirming the properties of the parabola.