Translate Function: Find G(x) From F(x) = X^2

In mathematics, understanding transformations of functions is crucial. One common transformation is a vertical translation, where a function is shifted up or down along the y-axis. In this article, we'll explore how to find a new function, g(x), that represents a vertical translation of the function f(x) = x^2 by 3 units upward. We will express g(x) in the vertex form a(x - h)^2 + k, where a, h, and k are integers. This form is particularly useful for identifying the vertex of the parabola and understanding its orientation.

Understanding Vertical Translations

A vertical translation involves shifting a function's graph up or down without changing its shape. When a function f(x) is translated vertically by k units, the new function g(x) is given by:

g(x) = f(x) + k

If k is positive, the graph shifts upward. If k is negative, the graph shifts downward. This simple addition or subtraction of a constant is a fundamental concept in function transformations and is widely applicable in various mathematical contexts. For example, in physics, understanding vertical translations can help analyze the potential energy of a system. In computer graphics, it can be used to move objects up or down on the screen. Therefore, mastering this concept is essential for anyone studying mathematics or related fields.

The impact of this translation is significant. It alters the range of the function while preserving its domain. For instance, if f(x) has a minimum value, translating it upward by k units will shift the minimum value up by the same amount. This understanding is particularly useful when analyzing the behavior of functions and their graphical representations.

Finding g(x)

Given the function f(x) = x^2, we want to find g(x), which is the translation of f(x) by 3 units upward. Using the formula for vertical translation, we have:

g(x) = f(x) + 3

Substitute f(x) = x^2 into the equation:

g(x) = x^2 + 3

Now, we need to express g(x) in the form a(x - h)^2 + k. In this case, we can rewrite g(x) as:

g(x) = 1(x - 0)^2 + 3

Here, a = 1, h = 0, and k = 3. This form clearly shows that the vertex of the parabola is at the point (0, 3), and the parabola opens upwards since a is positive. The value of understanding how to rewrite functions in vertex form is immense, as it allows for quick identification of key features of the graph, such as the vertex, axis of symmetry, and direction of opening.

Expressing g(x) in Vertex Form

We have already found g(x) and expressed it in the required form:

g(x) = 1(x - 0)^2 + 3

This form confirms that a = 1, h = 0, and k = 3. The vertex of the parabola is at (0, 3), which is the point where the parabola changes direction. The axis of symmetry is the vertical line x = 0, which passes through the vertex. Since a > 0, the parabola opens upwards, indicating that the vertex is the minimum point of the function. This detailed analysis provides a complete understanding of the transformed function and its graphical representation.

The vertex form a(x - h)^2 + k is particularly useful because it directly reveals the vertex of the parabola, which is at the point (h, k). In our case, h = 0 and k = 3, so the vertex is at (0, 3). This means that the original parabola f(x) = x^2, which has its vertex at (0, 0), has been shifted 3 units upward. The coefficient a determines the direction and steepness of the parabola. If a is positive, the parabola opens upward, and if a is negative, it opens downward. The larger the absolute value of a, the steeper the parabola.

Conclusion

In summary, the function g(x), which is the translation of f(x) = x^2 by 3 units upward, is given by:

g(x) = 1(x - 0)^2 + 3

This is in the form a(x - h)^2 + k, where a = 1, h = 0, and k = 3. Understanding vertical translations and expressing functions in vertex form are fundamental skills in mathematics, with applications in various fields. By mastering these concepts, you can analyze and manipulate functions more effectively. Remember, vertical translations involve shifting the graph of a function up or down by adding or subtracting a constant, while the vertex form provides valuable information about the parabola's vertex, axis of symmetry, and direction of opening. Keep practicing these transformations to enhance your mathematical skills.

For further learning and a deeper dive into function transformations, consider exploring resources like Khan Academy's Transformations of Functions.