Graphing Y=(x-3)^2+1 A Comprehensive Guide

In the realm of mathematics, understanding graphs of functions is a fundamental skill. It allows us to visualize the behavior of equations and gain deeper insights into their properties. This article will delve into the specifics of graphing the quadratic function $y=(x-3)^2+1$. We will explore the concepts behind this equation, the transformations involved, and how to accurately represent it graphically. By the end of this discussion, you will have a solid grasp of how to approach similar graphing problems and interpret the visual representation of quadratic functions. This exploration is not only crucial for academic success in mathematics but also for various real-world applications where visualizing relationships between variables is essential.

1. Decoding the Equation: $y=(x-3)^2+1$

To effectively graph the equation $y=(x-3)^2+1$, we first need to understand the different components of this quadratic function. This equation is in vertex form, which is a particularly useful representation for identifying key features of the parabola. The general form of a quadratic equation in vertex form is $y = a(x-h)^2 + k$, where $(h, k)$ represents the vertex of the parabola and 'a' determines the direction and stretch of the parabola. In our case, $y=(x-3)^2+1$, we can clearly see that $a = 1$, $h = 3$, and $k = 1$. The value of 'a' being 1 indicates that the parabola opens upwards and has a standard width, meaning it's neither stretched nor compressed vertically compared to the basic parabola $y=x^2$. The vertex $(h, k)$ is a critical point because it represents the minimum or maximum value of the function, depending on the sign of 'a'. For our equation, the vertex is $(3, 1)$, which means the lowest point on the parabola is at $x = 3$ and $y = 1$. This vertex serves as the anchor point around which the rest of the parabola is shaped. Understanding these components is crucial because it allows us to quickly sketch the graph without having to plot numerous points. By recognizing the vertex form and identifying the parameters, we can efficiently determine the parabola's position and orientation on the coordinate plane. This knowledge not only simplifies the graphing process but also enhances our ability to analyze and interpret quadratic functions in various mathematical contexts.

2. Transformations: Shifting the Parabola

Understanding the transformations applied to the basic parabola $y=x^2$ is key to graphing $y=(x-3)^2+1$. The given equation represents a transformation of the parent function, which is the simplest form of a parabola, $y=x^2$. The transformations involved are horizontal and vertical shifts. The term $(x-3)$ inside the square indicates a horizontal shift. Specifically, it shifts the parabola 3 units to the right along the x-axis. This might seem counterintuitive, but it's important to remember that the shift is in the opposite direction of the sign inside the parentheses. So, $(x-3)$ means a shift to the right. The $+1$ outside the squared term represents a vertical shift. This shifts the parabola 1 unit upwards along the y-axis. In essence, these transformations take the basic parabola $y=x^2$, move it 3 units to the right, and then 1 unit up. The vertex of the basic parabola, which is at $(0,0)$, is therefore shifted to $(3,1)$, which becomes the vertex of the transformed parabola. Recognizing these transformations allows us to visualize the graph without needing to plot individual points. We start with the basic parabola, then apply the horizontal shift to position it correctly along the x-axis, and finally apply the vertical shift to finalize its position on the coordinate plane. This understanding of transformations is not only applicable to quadratic functions but also to other types of functions, making it a valuable tool in graphical analysis. By mastering transformations, we can quickly and accurately sketch graphs and understand how different algebraic manipulations affect the shape and position of functions.

3. Plotting Key Points and Sketching the Graph

With the vertex and transformations understood, we can now plot key points and sketch the graph of $y=(x-3)^2+1$. The vertex, which we identified as $(3, 1)$, is the most crucial point to plot as it serves as the base of the parabola. Since the parabola opens upwards (because the coefficient 'a' is positive), the vertex represents the minimum point of the function. To sketch the graph accurately, it's beneficial to plot a few additional points on either side of the vertex. We can choose x-values that are relatively close to 3 to keep the calculations manageable. For instance, let's choose $x = 2$ and $x = 4$. When $x = 2$, $y = (2-3)^2 + 1 = (-1)^2 + 1 = 2$. So, the point $(2, 2)$ is on the graph. When $x = 4$, $y = (4-3)^2 + 1 = (1)^2 + 1 = 2$. This gives us the point $(4, 2)$. Notice that the y-values are the same for $x = 2$ and $x = 4$, which is a characteristic of parabolas due to their symmetry about the vertical line passing through the vertex (the axis of symmetry). We can also find the y-intercept by setting $x = 0$. When $x = 0$, $y = (0-3)^2 + 1 = (-3)^2 + 1 = 10$. This gives us the y-intercept at $(0, 10)$. Plotting these points – the vertex $(3, 1)$, the points $(2, 2)$ and $(4, 2)$, and the y-intercept $(0, 10)$ – provides a good framework for sketching the parabola. We draw a smooth, U-shaped curve that passes through these points, ensuring that the curve is symmetrical about the vertical line $x = 3$ (the axis of symmetry). The resulting graph is a parabola that opens upwards, with its lowest point at the vertex $(3, 1)$. This method of plotting key points and using the understanding of transformations allows for a precise and efficient way to sketch quadratic functions. It combines algebraic understanding with visual representation, making it a powerful tool in mathematical analysis.

4. Key Features of the Graph

Analyzing the graph of $y=(x-3)^2+1$ reveals several key features that are characteristic of parabolas. Understanding these features not only helps in accurately sketching the graph but also in interpreting the behavior of the quadratic function it represents. The most prominent feature is the vertex, which, as we've established, is located at $(3, 1)$. The vertex is the point where the parabola changes direction; in this case, it's the minimum point because the parabola opens upwards. The axis of symmetry is another crucial feature. It's a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. For this graph, the axis of symmetry is the line $x = 3$. This symmetry means that for any point on the parabola, there's a corresponding point on the other side of the axis of symmetry at the same y-value. The y-intercept, which is the point where the graph intersects the y-axis, is another important feature. We found the y-intercept to be $(0, 10)$. This point provides information about the function's value when $x = 0$. Unlike the x-intercepts, which are the points where the graph intersects the x-axis, this parabola does not have any x-intercepts. This is because the vertex is above the x-axis, and the parabola opens upwards, so it never crosses the x-axis. The domain of this quadratic function is all real numbers, meaning the function is defined for any value of $x$. This is typical for parabolas. The range, however, is restricted. Since the vertex is the minimum point and the parabola opens upwards, the range is all y-values greater than or equal to the y-coordinate of the vertex. In this case, the range is $y \geq 1$. These key features – the vertex, axis of symmetry, y-intercept, x-intercepts (or lack thereof), domain, and range – provide a comprehensive understanding of the graph's behavior and the properties of the quadratic function. Analyzing these features is essential for both sketching the graph accurately and interpreting the function in various mathematical and real-world contexts.

5. Applications and Significance

The ability to graph and analyze equations like $y=(x-3)^2+1$ extends far beyond the classroom. Quadratic functions and their graphs, parabolas, have numerous applications and significant implications in various fields. In physics, parabolas describe the trajectory of projectiles, such as a ball thrown in the air. The equation we've analyzed could, for example, represent the height of a projectile over time, where the vertex would indicate the maximum height reached. Understanding the vertex and the shape of the parabola allows physicists to calculate the range, maximum height, and time of flight for projectiles. In engineering, parabolas are used in the design of bridges, arches, and satellite dishes. The parabolic shape of a satellite dish, for instance, allows it to focus incoming signals at a single point, the focus, which is a crucial property for signal reception. Similarly, parabolic arches in bridges provide structural stability and efficient load distribution. In economics, quadratic functions can model cost, revenue, and profit curves. The vertex of the parabola in these models can represent the point of maximum profit or minimum cost, providing valuable insights for business decision-making. In computer graphics, parabolas are used to create smooth curves and shapes in animations and computer-aided design (CAD) software. The mathematical properties of parabolas make them ideal for generating visually appealing and mathematically precise graphics. Beyond these specific examples, the ability to analyze graphs and functions is a fundamental skill in many STEM fields. It allows professionals to model real-world phenomena, make predictions, and solve complex problems. Understanding the behavior of functions through their graphical representation provides a powerful tool for analysis and decision-making. Therefore, mastering the concepts of graphing quadratic functions, like the one we've explored, is not just an academic exercise but a crucial skill for success in a wide range of disciplines.