Transformations: F(x) = X² To G(x) = (x-1)² + 2

by Alex Johnson 48 views

\nTransformations of functions are a fundamental concept in mathematics, allowing us to understand how the graph of a function changes when we alter its equation. In this article, we'll explore the transformation from the basic quadratic function f(x) = x² to the transformed function g(x) = (x-1)² + 2. We'll break down each component of the transformation, explaining how it affects the graph and providing a clear, intuitive understanding.

The Basic Quadratic Function: f(x) = x²

Before diving into the transformation, let's quickly review the basic quadratic function, f(x) = x². This function represents a parabola with its vertex at the origin (0, 0). The graph is symmetrical around the y-axis, and as x moves away from 0 in either direction, f(x) increases rapidly. This simple function serves as the foundation for understanding more complex quadratic functions through transformations.

Key characteristics of f(x) = x²:

  • Vertex at (0, 0)
  • Symmetrical about the y-axis
  • Opens upwards
  • The simplest form of a quadratic function

Understanding these basic characteristics is crucial, as they will be altered by the transformations we apply.

Introducing the Transformed Function: g(x) = (x-1)² + 2

Now let's examine the transformed function, g(x) = (x-1)² + 2. This function looks similar to f(x) = x², but with some key differences. The '(x-1)' inside the parentheses and the '+2' outside represent transformations that shift the graph of the original function. Our goal is to identify and describe these transformations precisely.

The general form of a transformed quadratic function is g(x) = a(x-h)² + k, where:

  • a represents a vertical stretch or compression (and reflection if negative).
  • h represents a horizontal translation.
  • k represents a vertical translation.

In our case, a = 1, h = 1, and k = 2. Understanding these parameters is key to describing the transformation. We will focus on h and k since a is 1 in this example.

Horizontal Translation: The (x - 1) Term

The term (x - 1) inside the parentheses represents a horizontal translation. Specifically, it shifts the graph of f(x) horizontally. The rule to remember is that (x - h) shifts the graph h units to the right if h is positive and h units to the left if h is negative.

In our function, g(x) = (x-1)² + 2, we have (x - 1), which means h = 1. Therefore, the graph of f(x) is translated 1 unit to the right. Think of it this way: to get the same y-value as f(x), you need to input a value that is 1 unit larger into g(x). For instance, to get g(x) = 0, you need to input x = 1 into g(x), while you input x = 0 into f(x).

To further clarify, consider a few points on the graph of f(x) = x² and how they shift in g(x) = (x-1)² + 2:

  • The point (0, 0) on f(x) shifts to (1, 0) before considering the vertical translation. You can see that to get the y value 0, the x value had to be shifted by 1 unit, i.e. from 0 to 1.
  • The point (1, 1) on f(x) shifts to (2, 1) before considering the vertical translation. You can see that to get the y value 1, the x value had to be shifted by 1 unit, i.e. from 1 to 2.

This consistent shift to the right is a direct consequence of the (x - 1) term. Understanding this horizontal translation is a crucial step in fully comprehending the transformation.

The horizontal translation is 1 unit to the right.

Vertical Translation: The + 2 Term

The term + 2 outside the parentheses represents a vertical translation. This shifts the entire graph of f(x) upwards or downwards. A positive value shifts the graph upwards, while a negative value shifts it downwards. In our function, g(x) = (x-1)² + 2, we have + 2, which means the graph of f(x) is translated 2 units upwards.

To illustrate this, consider what happens to the vertex of the parabola. The vertex of f(x) = x² is at (0, 0). After the horizontal translation, it moves to (1, 0). Then, the + 2 shifts it upwards to (1, 2). Thus, the vertex of g(x) is at (1, 2).

Similarly, every other point on the graph of f(x) is also shifted 2 units upwards. For example:

  • The point (1, 1) on f(x) becomes (2, 1) after the horizontal translation, and then (2, 3) after the vertical translation.
  • The point (-1, 1) on f(x) becomes (0, 1) after the horizontal translation, and then (0, 3) after the vertical translation.

This consistent shift upwards is due to the + 2 term. Combining this with the horizontal translation gives us the complete transformation.

The vertical translation is 2 units up.

Combining the Transformations

Now that we've analyzed each component separately, let's combine them to describe the overall transformation from f(x) = x² to g(x) = (x-1)² + 2. The graph of f(x) is translated 1 unit to the right and 2 units up. This means that every point on the original parabola is moved 1 unit to the right and 2 units upwards to create the new parabola.

The vertex of the original parabola at (0, 0) moves to (1, 2). The axis of symmetry, originally the y-axis, shifts to the vertical line x = 1. The overall shape of the parabola remains the same; it's simply shifted to a new location in the coordinate plane.

In summary, the transformation is a translation 1 unit to the right and 2 units up.

Visualizing the Transformation

To solidify your understanding, it's helpful to visualize the transformation. Imagine the graph of f(x) = x² as a parabola sitting at the origin. Now, grab that parabola and slide it 1 unit to the right. Then, lift it 2 units upwards. The resulting parabola is the graph of g(x) = (x-1)² + 2. Visualizing the transformation in this way can make it easier to remember and apply the concepts.

Conclusion

Understanding transformations of functions is a vital skill in mathematics. By breaking down complex functions into simpler components, we can analyze and describe how their graphs change. In the case of g(x) = (x-1)² + 2, we identified a horizontal translation of 1 unit to the right and a vertical translation of 2 units up, relative to the basic quadratic function f(x) = x². Mastering these transformations will allow you to confidently analyze and manipulate a wide range of functions.

For further exploration of function transformations, consider visiting Khan Academy's section on Transformations of Functions.