Understanding the Vertex of the Parabola: Why x 0 when y -1/2x^2 9

Posals of quadratic functions like the parabola y -1/2x^2 9 often lead to questions about the vertex and its characteristics. In this guide, we will elucidate the significance of the vertex and why the x-coordinate of the vertex for this particular parabola is x 0. We will also delve into the transformations involved and provide a step-by-step explanation of how to find the vertex.

Quadratic Functions and Their Forms

A quadratic function, in general, can be expressed as y ax^2 bx c, where a, b, c are constants and a ≠ 0. This is the standard form of a quadratic function. The parent function of the quadratic family, y x^2, serves as a basis for understanding the transformations and properties of more complex quadratic equations.

The Vertex Form of a Quadratic Function

In the vertex form of a quadratic function, y a(x - h)^2 k, the coordinates of the vertex are directly given by (h, k). The variable a influences the shape and orientation of the parabola. If a 0, the parabola opens upwards, and if a 0, it opens downwards. The variable h represents the horizontal shift, and k represents the vertical shift.

Transformations and Their Effects

To understand the given equation y -1/2x^2 9, let's break down the transformations involved:

Vertical Stretch/Compression: The coefficient a -1/2 indicates a vertical compression by a factor of 2 and a reflection of the graph over the x-axis. Horizontal Shift: Since h 0, there is no horizontal shift. Vertical Shift: The constant k 9 indicates a vertical shift upwards by 9 units.

Given these transformations, we can analyze the equation in more detail. Since the horizontal shift h 0, the vertex of the parabola in the parent function y x^2 remains at (0, 0), and then it is shifted vertically 9 units upwards.

Finding the Vertex

To find the vertex of y -1/2x^2 9, we can follow these steps:

Start with the general form: y -1/2x^2 9. Rewrite it in vertex form: y -1/2(x - 0)^2 9. From the vertex form, we can directly read off the vertex coordinates, which are (h, k) (0, 9). To verify, substitute x 0 into the original equation:

y -1/2(0)^2 9 9. Thus, the vertex is indeed at (0, 9).

Additional Insights

We can also understand why the x-coordinate of the vertex is 0 by using the vertex formula for a quadratic polynomial ax^2 bx c:

x -b/2a

In the given equation, a -1/2 and b 0. Substituting these values into the formula:

x -0 / -1/2 0

Therefore, the x-coordinate of the vertex of the parabola is 0.

Conclusion

The x-coordinate of the vertex for the parabola described by the equation y -1/2x^2 9 is 0 because there is no horizontal shift involved, and the horizontal shift for the parent function y x^2 is at x 0. This conclusion is further validated by both the vertex form and the vertex formula.

Visual Aid

For a more visual understanding, refer to the Desmos graph of the given parabola: Desmos graph.

Related Questions and Further Exploration

For further exploration, consider the following related questions:

How do transformations affect the graph of a quadratic function? What is the significance of the vertex in a quadratic function, and how does it reflect the behavior of the parabola? How can you use the vertex formula to find the vertex of any quadratic function?