Graphing The Parabola Y=(x-5)^2-1 A Step-by-Step Guide

Understanding Parabolas

When you need to graph a parabola, it's not just about plotting random points; it's about understanding the underlying structure and properties of the parabola itself. The equation $y = (x-5)^2 - 1$ represents a parabola, a U-shaped curve that is fundamental in mathematics and physics. This article provides a comprehensive guide on how to graph this specific parabola, ensuring a thorough understanding of the process and the concepts involved.

This guide focuses on graphing the parabola defined by the equation $y = (x-5)^2 - 1$. This equation is in vertex form, which is incredibly useful for quickly identifying key features of the parabola, such as its vertex, axis of symmetry, and direction of opening. Our goal is to plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. This approach will give us a clear and accurate representation of the parabola's shape and position on the coordinate plane. By the end of this guide, you'll not only be able to graph this specific parabola but also understand the principles behind graphing any parabola given in vertex form.

Identifying Key Features from Vertex Form

To effectively graph the parabola $y = (x-5)^2 - 1$, we first need to dissect the equation and identify its key features. The vertex form of a parabola's equation is given by $y = a(x-h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. The vertex is the turning point of the parabola; it is either the minimum or maximum point on the curve. The value of 'a' determines the direction the parabola opens (upward if $a > 0$, downward if $a < 0$) and how wide or narrow it is.

In our equation, $y = (x-5)^2 - 1$, we can directly identify the values of $h$, $k$, and $a$. By comparing our equation with the standard vertex form, we can see that $h = 5$, $k = -1$, and $a = 1$. This immediately tells us that the vertex of our parabola is at the point $(5, -1)$. The fact that $a = 1$ (a positive value) indicates that the parabola opens upwards, meaning the vertex is the minimum point on the curve. Furthermore, since $a = 1$, the parabola has a standard width, neither stretched nor compressed compared to the basic parabola $y = x^2$.

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. The equation of the axis of symmetry is given by $x = h$. In our case, the axis of symmetry is the vertical line $x = 5$. Understanding the axis of symmetry helps us plot points efficiently, as we know that for every point on one side of the axis, there is a corresponding point on the other side at the same vertical distance from the vertex. This symmetry simplifies the process of graphing, allowing us to calculate fewer points while still accurately representing the parabola's shape.

Calculating Points to Plot

Now that we've identified the vertex $(5, -1)$ and the direction of opening (upwards), the next step is to calculate additional points to plot on the graph. We need two points to the left of the vertex and two points to the right. Choosing points symmetrically around the vertex helps maintain accuracy and makes the graphing process more efficient. It's wise to select integer values for x that are close to the vertex's x-coordinate to keep the calculations manageable and the plotted points within a reasonable range on the graph.

Let's start by selecting $x$-values that are two units to the left and right of the vertex's $x$-coordinate, which is 5. This gives us $x = 3$ and $x = 7$. We'll also choose $x$-values that are one unit away from the vertex, $x = 4$ and $x = 6$. These choices provide a good spread of points around the vertex, allowing us to capture the curve's shape accurately. Now, we'll substitute each of these $x$-values into the equation $y = (x-5)^2 - 1$ to find the corresponding $y$-values.

For $x = 3$, we have $y = (3-5)^2 - 1 = (-2)^2 - 1 = 4 - 1 = 3$. So, the point is $(3, 3)$. For $x = 4$, we have $y = (4-5)^2 - 1 = (-1)^2 - 1 = 1 - 1 = 0$. So, the point is $(4, 0)$. For $x = 6$, we have $y = (6-5)^2 - 1 = (1)^2 - 1 = 1 - 1 = 0$. So, the point is $(6, 0)$. For $x = 7$, we have $y = (7-5)^2 - 1 = (2)^2 - 1 = 4 - 1 = 3$. So, the point is $(7, 3)$.

Thus, we have the following five points to plot: the vertex $(5, -1)$, two points to the left $(3, 3)$ and $(4, 0)$, and two points to the right $(6, 0)$ and $(7, 3)$. These points will give us a clear picture of the parabola's shape and position on the coordinate plane.

Plotting the Points and Graphing the Parabola

With our five points calculated—the vertex $(5, -1)$, $(3, 3)$, $(4, 0)$, $(6, 0)$, and $(7, 3)$—we're ready to plot them on the coordinate plane. Start by drawing the $x$ and $y$ axes, ensuring you have enough space to accommodate the points we've calculated. The x-values range from 3 to 7, and the y-values range from -1 to 3, so your axes should cover these ranges adequately.

Now, carefully plot each point. The vertex $(5, -1)$ is the lowest point on the graph, as the parabola opens upwards. The points $(4, 0)$ and $(6, 0)$ are equidistant from the vertex and lie on the x-axis, indicating the parabola's x-intercepts. Similarly, the points $(3, 3)$ and $(7, 3)$ are symmetrically positioned relative to the vertex and higher up on the graph. Once you've plotted all five points, you should see a clear U-shaped pattern emerging.

To graph the parabola, draw a smooth curve that passes through all the plotted points. Remember that the parabola is symmetrical around its axis of symmetry ($x = 5$ in this case), so the curve should mirror itself on either side of the vertex. The curve should extend upwards from the vertex, gradually widening as it moves away from the vertex. Avoid drawing sharp corners or straight lines; the parabola should be a smooth, continuous curve. As you draw, keep the overall shape and symmetry of the parabola in mind, ensuring that your graph accurately represents the equation $y = (x-5)^2 - 1$.

Understanding the Graph

Once the parabola is graphed, take a moment to analyze the graph and connect it back to the equation $y = (x-5)^2 - 1$. The graph visually confirms the properties we deduced earlier: the vertex is indeed at $(5, -1)$, the parabola opens upwards, and the axis of symmetry is the line $x = 5$. The graph also shows us the behavior of the function; as we move away from the vertex along the x-axis, the y-values increase, indicating the parabola's upward trend.

Moreover, we can identify the x-intercepts from the graph, which are the points where the parabola crosses the x-axis. In this case, the x-intercepts are at $(4, 0)$ and $(6, 0)$, corresponding to the points we calculated earlier. These intercepts are the solutions to the equation $(x-5)^2 - 1 = 0$, and they represent the values of x for which the parabola's y-value is zero. The y-intercept, which is the point where the parabola crosses the y-axis, can also be determined from the graph, although it's not one of the points we explicitly plotted. To find the y-intercept algebraically, you would set $x = 0$ in the equation and solve for $y$.

The graph also illustrates the concept of transformations. The parabola $y = (x-5)^2 - 1$ is a transformation of the basic parabola $y = x^2$. The $(x-5)$ term represents a horizontal shift of 5 units to the right, and the $-1$ represents a vertical shift of 1 unit downwards. These transformations move the vertex from $(0, 0)$ for the basic parabola to $(5, -1)$ for our given equation. Understanding these transformations allows us to quickly visualize and graph parabolas without having to plot numerous points.

Conclusion

Graphing the parabola $y = (x-5)^2 - 1$ involves a systematic approach of identifying key features from the equation, calculating points to plot, and drawing a smooth curve through those points. By understanding the vertex form of a parabola, we can quickly determine the vertex, axis of symmetry, and direction of opening. Calculating additional points, especially those symmetrically positioned around the vertex, ensures an accurate representation of the parabola's shape.

Plotting these points and drawing a smooth curve results in a visual representation of the quadratic function, allowing us to understand its behavior and properties. The graph confirms our initial analysis and provides insights into the function's intercepts, symmetry, and transformations. This process not only enables us to graph a specific parabola but also provides a foundation for understanding and graphing other quadratic functions.

By following this step-by-step guide, you can confidently graph parabolas and appreciate their significance in mathematics and real-world applications. Whether you're solving quadratic equations, modeling projectile motion, or designing parabolic reflectors, the ability to graph and understand parabolas is an invaluable skill.