Shifting Quadratic Functions A Step-by-Step Guide

In the realm of function transformations, understanding horizontal shifts is crucial for manipulating graphs and equations. This article delves into the concept of horizontal shifts, specifically focusing on quadratic functions. We will explore how shifting a function to the left or right affects its equation and graph. Mastering these transformations is essential for a deeper understanding of mathematical functions and their applications.

The question at hand involves a quadratic function, $f(x) = x^2 - 4x + 4$, and its transformation. The function is shifted 3 units to the left, resulting in a new function, $g(x)$. Our goal is to determine the equation for $g(x)$. This problem highlights the core principle of horizontal shifts: replacing x with (x + c) shifts the graph c units to the left, while replacing x with (x - c) shifts the graph c units to the right. This seemingly counterintuitive behavior is a key concept to grasp when working with function transformations.

To solve this problem, we need to apply the principle of horizontal shifts to the given function. We will replace x in the original function with (x + 3), as shifting 3 units to the left corresponds to adding 3 to the x value inside the function. This substitution will give us the equation for $g(x)$. We will then simplify the resulting expression to match one of the given options. This process involves algebraic manipulation and a clear understanding of function notation.

To find the new function $g(x)$ after shifting $f(x)$ three units to the left, we need to apply the horizontal shift transformation. In general, if we have a function $f(x)$ and we want to shift it c units to the left, we replace every instance of x in the function with (x + c). Conversely, if we want to shift it c units to the right, we replace x with (x - c). In our case, we have the function $f(x) = x^2 - 4x + 4$ and we want to shift it 3 units to the left. This means we need to replace x with (x + 3) in the expression for $f(x)$.

Applying this transformation, we get:

$g(x) = f(x + 3) = (x + 3)^2 - 4(x + 3) + 4$

This expression represents the function $g(x)$ after the horizontal shift. Now, we need to simplify this expression to see if it matches any of the given options. Expanding the terms, we have:

$g(x) = (x^2 + 6x + 9) - 4(x + 3) + 4$

Next, distribute the -4:

$g(x) = x^2 + 6x + 9 - 4x - 12 + 4$

Finally, combine like terms:

$g(x) = x^2 + (6x - 4x) + (9 - 12 + 4)$

$g(x) = x^2 + 2x + 1$

Now, let's compare this simplified expression with the given options to find the correct answer. Option C, $g(x) = (x + 3)^2 - 4(x + 3) + 4$, directly represents the substitution we made to shift the function 3 units to the left. This confirms that option C is the correct answer.

Now that we've derived the correct equation for $g(x)$, let's analyze the given options to understand why the other options are incorrect. This will reinforce our understanding of horizontal shifts and function transformations.

• Option A: $g(x) = (x^2 - 4x + 4) - 3$

  This option represents a vertical shift of the original function downward by 3 units. A vertical shift affects the y-values of the function, not the x-values. To shift a function vertically, we add or subtract a constant outside the function's argument. Therefore, this option is incorrect as it doesn't represent a horizontal shift.

• Option B: $g(x) = (x - 3)^2 - 4(x - 3) + 4$

  This option represents a horizontal shift of the original function to the right by 3 units. Replacing x with (x - 3) corresponds to a shift to the right, not the left. This is the opposite direction of the shift we were asked to perform. Thus, this option is incorrect.

• Option C: $g(x) = (x + 3)^2 - 4(x + 3) + 4$

  This is the correct option. As we derived earlier, this equation represents the function $f(x)$ shifted 3 units to the left. Replacing x with (x + 3) is the correct transformation for a leftward horizontal shift.

• Option D: $g(x) = (x^2 - 4x + 4) + 3$

  Similar to option A, this option represents a vertical shift of the original function upward by 3 units. Adding a constant outside the function's argument results in a vertical shift. This option is incorrect because it does not represent a horizontal shift.

By analyzing each option, we can clearly see why only option C correctly represents the horizontal shift of the function 3 units to the left. This reinforces the importance of understanding the difference between horizontal and vertical shifts and how they are represented in function notation.

To further solidify our understanding, let's visualize the shift graphically. The original function, $f(x) = x^2 - 4x + 4$, can be rewritten as $f(x) = (x - 2)^2$. This is a parabola with its vertex at the point (2, 0). Shifting this parabola 3 units to the left means that the vertex will also shift 3 units to the left. The new vertex will be at the point (2 - 3, 0) = (-1, 0).

The shifted function, $g(x) = (x + 3)^2 - 4(x + 3) + 4$, simplifies to $g(x) = x^2 + 2x + 1$, which can be further rewritten as $g(x) = (x + 1)^2$. This is also a parabola, but its vertex is at the point (-1, 0), as we predicted. The graph of $g(x)$ is the same shape as the graph of $f(x)$, but it is shifted 3 units to the left.

Visualizing the shift helps to confirm our algebraic solution. We can see that the graph of $g(x)$ is indeed the graph of $f(x)$ moved 3 units to the left. This graphical representation provides a concrete understanding of the transformation and its effect on the function.

This problem illustrates several key concepts related to function transformations, particularly horizontal shifts:

• Horizontal Shifts: Replacing x with (x + c) shifts the graph c units to the left, while replacing x with (x - c) shifts the graph c units to the right. This counterintuitive behavior is a fundamental aspect of horizontal transformations.
• Vertical Shifts: Adding or subtracting a constant outside the function's argument results in a vertical shift. Adding a constant shifts the graph upward, while subtracting a constant shifts the graph downward.
• Function Notation: Understanding function notation is crucial for correctly applying transformations. The notation f(x + c) indicates a horizontal shift, while f(x) + c indicates a vertical shift.
• Graphical Representation: Visualizing the graphs of functions and their transformations provides a powerful way to understand the effects of these transformations. The graphical representation confirms our algebraic solutions and deepens our understanding.

By mastering these concepts, you can confidently tackle problems involving function transformations and gain a deeper appreciation for the behavior of mathematical functions.

In conclusion, the function $g(x)$, created by shifting $f(x) = x^2 - 4x + 4$ three units to the left, is represented by the equation $g(x) = (x + 3)^2 - 4(x + 3) + 4$. This problem highlights the critical concept of horizontal shifts in function transformations. By replacing x with (x + 3), we effectively shifted the graph of the function 3 units to the left. Understanding horizontal and vertical shifts, along with their corresponding algebraic representations, is essential for manipulating and analyzing functions. Visualizing these transformations graphically provides a deeper understanding and reinforces the algebraic solutions. This exercise demonstrates the importance of function notation and its role in accurately representing transformations. Mastering these concepts allows for a more profound comprehension of mathematical functions and their behavior.

By carefully applying the principles of horizontal shifts and verifying the solution through algebraic manipulation and graphical visualization, we have successfully determined the equation for the shifted function. This problem serves as a valuable example for understanding and applying function transformations.