Vertex At Origin: Identifying Functions With Vertex (0,0)

Determining which function has a vertex at the origin, (0, 0), is a fundamental concept in understanding quadratic functions and their graphical representations. The vertex of a parabola, which is the graph of a quadratic function, represents either the minimum or maximum point of the function. When the vertex is located at the origin, it signifies specific characteristics of the function's equation. Let's explore how to identify such functions by examining their equations and transformations. This article will walk you through the process of identifying functions with a vertex at the origin, focusing on the key properties and transformations of quadratic functions. Understanding this concept is crucial for various applications in mathematics, physics, and engineering, where parabolic trajectories and optimization problems are frequently encountered.

Understanding Vertex Form

To effectively identify functions with a vertex at the origin, a solid understanding of the vertex form of a quadratic equation is essential. The vertex form is given by:

f(x) = a(x - h)^2 + k

Where:

• (h, k) represents the coordinates of the vertex of the parabola.
• a determines the direction and steepness of the parabola. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The absolute value of a indicates how stretched or compressed the parabola is compared to the basic parabola f(x) = x^2.

The vertex form is incredibly useful because it directly reveals the vertex of the parabola. By simply looking at the values of h and k, we can pinpoint the vertex's location on the coordinate plane. This makes it much easier to analyze and compare different quadratic functions.

For a function to have its vertex at the origin (0, 0), both h and k must be equal to 0. This simplifies the vertex form equation to:

f(x) = ax^2

This simplified form is the key to recognizing functions with vertices at the origin. Any quadratic function that can be written in this form, where there are no constant terms or linear terms (terms with just x), will have its vertex at (0, 0). The coefficient a only affects the parabola's direction and how wide or narrow it is, but it does not shift the vertex from the origin. For example, f(x) = 2x^2, f(x) = -0.5x^2, and f(x) = -x^2 all have vertices at the origin because they fit this form.

Analyzing the Given Options

Now, let's delve into the options provided and determine which function has a vertex at the origin by applying our understanding of the vertex form. We'll examine each option individually, transforming them if necessary to see if they fit the form f(x) = ax^2.

Option A: f(x) = (x + 4)^2

To analyze this function, we need to expand the squared term:

f(x) = (x + 4)(x + 4)

f(x) = x^2 + 4x + 4x + 16

f(x) = x^2 + 8x + 16

This is a quadratic function in the standard form f(x) = ax^2 + bx + c, where a = 1, b = 8, and c = 16. To find the vertex, we can use the formula for the x-coordinate of the vertex, h = -b / 2a:

h = -8 / (2 * 1) = -4

Then, we find the y-coordinate, k, by substituting h back into the function:

k = f(-4) = (-4)^2 + 8(-4) + 16 = 16 - 32 + 16 = 0

Thus, the vertex of this parabola is at (-4, 0). Since the vertex is not at (0, 0), Option A is not the correct answer. The presence of the 8x term and the constant term 16 indicates a horizontal shift of the parabola away from the origin.

Option B: f(x) = x(x - 4)

Let's expand this function as well:

f(x) = x^2 - 4x

This is another quadratic function in standard form, where a = 1, b = -4, and c = 0. Again, we find the x-coordinate of the vertex:

h = -(-4) / (2 * 1) = 4 / 2 = 2

Now, we find the y-coordinate:

k = f(2) = (2)^2 - 4(2) = 4 - 8 = -4

Therefore, the vertex of this parabola is at (2, -4), which is not the origin. The presence of the -4x term shifts the vertex away from the origin, making Option B incorrect.

Option C: f(x) = (x - 4)(x + 4)

Expanding this function gives us:

f(x) = x^2 + 4x - 4x - 16

f(x) = x^2 - 16

This function is in the form f(x) = ax^2 + c, where a = 1 and c = -16. To find the vertex, we can recognize that this is a vertical shift of the basic parabola f(x) = x^2. The absence of a bx term indicates that the vertex will lie on the y-axis. The x-coordinate of the vertex is h = -b / 2a = 0, and the y-coordinate is k = f(0) = (0)^2 - 16 = -16. Thus, the vertex is at (0, -16), which is not the origin. The constant term -16 shifts the parabola downward, away from the origin.

Option D: f(x) = -x^2

This function is already in the form f(x) = ax^2, where a = -1. There are no additional terms, meaning h = 0 and k = 0. Therefore, the vertex of this parabola is at (0, 0), the origin. The negative sign simply indicates that the parabola opens downward, but it does not affect the vertex's location.

Conclusion

After analyzing all the options, we can confidently conclude that Option D, f(x) = -x^2, is the function with a vertex at the origin (0, 0). This function fits the form f(x) = ax^2, which is the key characteristic of quadratic functions with vertices at the origin.

Understanding the vertex form of a quadratic equation and how different terms affect the parabola's position and shape is crucial for solving these types of problems. By recognizing the basic form f(x) = ax^2, you can quickly identify functions with vertices at the origin. Remember, the coefficients and constants in the equation dictate the parabola's transformations, such as shifts and stretches. Mastering these concepts will not only help you in mathematics but also in various fields where quadratic functions are applied.

For further exploration of quadratic functions and their properties, consider visiting trusted resources like Khan Academy's section on quadratic functions.