Vertical Translation Of Quadratic Functions Analyzing The Shift From Y 2x2 To Y 2x2 5

This article delves into the concept of vertical translations of quadratic functions, specifically focusing on how to interpret the change from the graph of $y = 2x^2$ to the graph of $y = 2x^2 + 5$. We will explore the fundamental principles behind these transformations, providing a comprehensive understanding of the effect of adding a constant to a quadratic function. By the end of this discussion, you will be able to confidently identify and describe vertical shifts in quadratic graphs, enhancing your ability to analyze and manipulate these functions.

The question at hand asks us to determine the correct phrase that describes the transformation from the graph of the function $y = 2x^2$ to the graph of the function $y = 2x^2 + 5$. This falls under the category of graph transformations, a crucial concept in mathematics, particularly in algebra and calculus. Graph transformations involve altering the graph of a function while maintaining its basic shape. There are several types of transformations, including translations (shifts), reflections, stretches, and compressions. In this case, we are specifically dealing with a vertical translation, which means the graph is being shifted up or down along the y-axis. Let's break down the problem step by step to arrive at the correct answer.

Understanding the Parent Function: $y = 2x^2$

To begin, it's important to understand the parent function, which in this case is $y = 2x^2$. This is a quadratic function, meaning it has the general form $y = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants. In our parent function, $a = 2$, $b = 0$, and $c = 0$. The coefficient 'a' determines the vertical stretch or compression of the parabola. Since $a = 2$, the parabola will be narrower than the basic parabola $y = x^2$. The vertex of this parabola, which is the point where it changes direction, is at the origin (0, 0). The parabola opens upwards because the coefficient 'a' is positive. Understanding the basic shape and key features of the parent function is crucial for analyzing transformations.

The parabola $y = 2x^2$ is a fundamental quadratic function that serves as the base for our transformation. The graph of $y = 2x^2$ is a U-shaped curve, symmetrical about the y-axis, with its vertex at the origin (0,0). The coefficient 2 in front of the $x^2$ term indicates a vertical stretch, meaning the parabola is narrower compared to the standard parabola $y = x^2$. To visualize this, consider a few points on the graph. When $x = 0$, $y = 2(0)^2 = 0$. When $x = 1$, $y = 2(1)^2 = 2$. When $x = -1$, $y = 2(-1)^2 = 2$. These points help us sketch the basic shape of the parabola. Now that we have a solid understanding of the parent function, we can analyze the transformation introduced in the second equation.

Analyzing the Transformed Function: $y = 2x^2 + 5$

Now, let's examine the transformed function: $y = 2x^2 + 5$. Notice that this function is identical to the parent function $y = 2x^2$, except for the addition of the constant term '+ 5'. This addition is what causes a vertical translation. In general, adding a constant 'k' to a function, such as changing $f(x)$ to $f(x) + k$, results in a vertical shift of the graph. If 'k' is positive, the graph shifts upwards by 'k' units. If 'k' is negative, the graph shifts downwards by 'k' units. In our case, $k = 5$, which is positive. Therefore, the graph of $y = 2x^2 + 5$ is the same as the graph of $y = 2x^2$ shifted 5 units upwards.

The addition of the constant term +5 to the function $y = 2x^2$ is the key to understanding the vertical translation. This constant term directly affects the y-coordinate of every point on the graph. For any given x-value, the corresponding y-value on the graph of $y = 2x^2 + 5$ will be 5 units greater than the y-value on the graph of $y = 2x^2$. This means that the entire graph of the parent function is lifted vertically by 5 units. The vertex of the transformed parabola will now be at the point (0, 5), as the original vertex (0, 0) has been shifted upwards by 5 units. To further illustrate this, consider the points we discussed earlier. On the transformed graph, when $x = 0$, $y = 2(0)^2 + 5 = 5$. When $x = 1$, $y = 2(1)^2 + 5 = 7$. When $x = -1$, $y = 2(-1)^2 + 5 = 7$. This confirms that the parabola has been shifted upwards, maintaining its shape but changing its position on the coordinate plane.

Identifying the Correct Translation

Based on our analysis, we can now confidently identify the correct phrase that describes the translation. Since the graph of $y = 2x^2 + 5$ is obtained by shifting the graph of $y = 2x^2$ upwards by 5 units, the correct answer is:

• A. 5 units up

The other options are incorrect. Shifting the graph down would correspond to subtracting a constant, shifting it right or left would involve changes within the x-term (e.g., $y = 2(x - 5)^2$ would shift the graph 5 units to the right). Understanding these distinctions is crucial for correctly interpreting graph transformations.

To solidify our understanding, let's briefly discuss why the other options are incorrect. Option B, “5 units down,” would be represented by the equation $y = 2x^2 - 5$. Option C, “5 units right,” would involve a horizontal shift and would be represented by an equation of the form $y = 2(x - 5)^2$. Option D, “5 units left,” would similarly involve a horizontal shift and would be represented by an equation of the form $y = 2(x + 5)^2$. By recognizing the specific form of the equation that results in a vertical shift, we can accurately identify the correct transformation. The key takeaway is that adding a constant outside the squared term results in a vertical translation, and the sign of the constant determines the direction of the shift.

Conclusion

In summary, the addition of a constant to a quadratic function results in a vertical translation of its graph. Adding a positive constant shifts the graph upwards, while adding a negative constant shifts it downwards. In the given problem, the graph of $y = 2x^2 + 5$ is obtained by shifting the graph of $y = 2x^2$ five units upwards. Understanding these fundamental concepts of graph transformations is essential for success in algebra and calculus. By analyzing the equation and identifying the changes made to the parent function, we can accurately describe the resulting transformations. This exercise highlights the importance of recognizing the role of constants in altering the position of a graph on the coordinate plane.

This analysis provides a clear understanding of vertical translations in quadratic functions. By identifying the constant term added to the function, we can accurately determine the magnitude and direction of the shift. This knowledge is fundamental for working with more complex graph transformations and function manipulations. Remember, the key is to recognize the relationship between the equation and the corresponding changes in the graph's position. Through practice and careful analysis, you can master the art of interpreting and applying graph transformations.

By understanding the relationship between the equation and the graph, you can accurately predict and describe transformations. This skill is not only crucial for mathematical problem-solving but also provides a deeper understanding of the nature of functions and their graphical representations. Keep practicing and exploring different types of transformations to further enhance your mathematical intuition and problem-solving abilities.