Identifying The Graph Of Y=x²-15x+54 A Comprehensive Guide
Unlocking the secrets of quadratic equations can feel like deciphering a hidden code. In the realm of mathematics, a quadratic equation, with its signature squared term, paints a unique curve on the graph known as a parabola. Today, we will embark on a journey to identify the graph of the quadratic equation y = x² - 15x + 54. Understanding the key elements of a quadratic equation empowers us to visualize its graphical representation without even plotting a single point. We'll delve into the significance of the coefficients, explore the concept of roots, and analyze the vertex, the parabola's turning point, to confidently match the equation to its corresponding graph. The ability to connect an algebraic equation with its visual counterpart is a cornerstone of mathematical literacy, providing a deeper understanding of the relationship between numbers and shapes. Let's begin our exploration by dissecting the equation and extracting the information it holds within.
Understanding the Quadratic Form
In order to identify the graph of y = x² - 15x + 54, we must first recognize the standard form of a quadratic equation, which is y = ax² + bx + c. In this general form, 'a', 'b', and 'c' are constants that dictate the shape and position of the parabola. The coefficient 'a' plays a crucial role in determining the parabola's orientation. If 'a' is positive, the parabola opens upwards, resembling a 'U' shape. Conversely, if 'a' is negative, the parabola opens downwards, forming an inverted 'U' shape. In our specific equation, y = x² - 15x + 54, the coefficient 'a' is 1, which is positive. This tells us immediately that the parabola will open upwards. This is a crucial first step in narrowing down the possible graphs. Next, the coefficients 'b' and 'c', along with 'a', collectively influence the parabola's position on the coordinate plane. The 'c' term specifically represents the y-intercept, the point where the parabola intersects the y-axis. In our equation, 'c' is 54, meaning the parabola will cross the y-axis at the point (0, 54). This piece of information is invaluable as it eliminates any graphs that do not pass through this specific point. By carefully analyzing these coefficients, we are gaining a comprehensive understanding of the parabola's basic characteristics before we even consider the more intricate aspects like the roots or the vertex. The power of understanding the quadratic form lies in its ability to quickly provide key insights into the graph's behavior.
Finding the Roots (x-intercepts)
Finding the roots, also known as x-intercepts, of the quadratic equation y = x² - 15x + 54 is a critical step in identifying its graph. The roots are the points where the parabola intersects the x-axis, which means the y-value is zero at these points. To find the roots, we set the equation equal to zero: x² - 15x + 54 = 0. This equation can be solved using several methods, including factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach. We need to find two numbers that multiply to 54 and add up to -15. These numbers are -6 and -9. Therefore, we can factor the equation as follows: (x - 6)(x - 9) = 0. Setting each factor equal to zero, we get x - 6 = 0 and x - 9 = 0. Solving for x, we find the roots to be x = 6 and x = 9. These roots tell us that the parabola intersects the x-axis at the points (6, 0) and (9, 0). This information is invaluable for identifying the correct graph, as it provides two specific points that the parabola must pass through. By knowing the roots, we can eliminate any graphs that do not cross the x-axis at these points. The roots, along with the y-intercept, provide a significant framework for visualizing the parabola's position and shape on the coordinate plane. The ability to efficiently find the roots of a quadratic equation is a fundamental skill in understanding its graphical representation.
Locating the Vertex: The Turning Point
The vertex of the parabola defined by the equation y = x² - 15x + 54 is a crucial point to identify as it represents the turning point of the curve. This point is either the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards. Since we've already established that our parabola opens upwards (because the coefficient of x² is positive), the vertex will be the minimum point. To find the vertex, we can use the formula x = -b / 2a, where 'a' and 'b' are the coefficients in the standard quadratic form y = ax² + bx + c. In our equation, a = 1 and b = -15, so the x-coordinate of the vertex is x = -(-15) / (2 * 1) = 15 / 2 = 7.5. This means the vertex lies on the vertical line x = 7.5. To find the y-coordinate of the vertex, we substitute this x-value back into the original equation: y = (7.5)² - 15(7.5) + 54 = 56.25 - 112.5 + 54 = -2.25. Therefore, the vertex of the parabola is the point (7.5, -2.25). This is a critical piece of information because it tells us the lowest point on the graph and its location in the coordinate plane. Knowing the vertex allows us to visualize the parabola's symmetry around the vertical line passing through the vertex. It also helps us to eliminate graphs that have a vertex in a different location. The vertex, along with the roots and the y-intercept, paints a comprehensive picture of the parabola's position and shape, making it much easier to identify the correct graph.
Putting It All Together: Identifying the Graph
Having dissected the equation y = x² - 15x + 54, we now possess all the necessary clues to confidently identify its graph. We know the parabola opens upwards because the coefficient of x² is positive. We've determined the roots to be x = 6 and x = 9, indicating the parabola intersects the x-axis at (6, 0) and (9, 0). The y-intercept is 54, meaning the parabola crosses the y-axis at (0, 54). And finally, we've pinpointed the vertex at (7.5, -2.25), the minimum point of the curve. Now, we must look for a graph that embodies all these characteristics. The graph should be a U-shaped parabola opening upwards. It must pass through the points (6, 0), (9, 0), and (0, 54). The lowest point on the graph should be at (7.5, -2.25). By carefully examining potential graphs and comparing them to these key features, we can eliminate any that don't match. The process of elimination, guided by our understanding of the equation's properties, will lead us to the correct graph. This exercise demonstrates the power of connecting algebraic equations with their visual representations. By understanding the significance of coefficients, roots, y-intercepts, and the vertex, we can confidently navigate the world of quadratic equations and their graphical counterparts. The ability to visualize equations is a fundamental skill in mathematics, and this example showcases how a systematic approach can unlock the mysteries of parabolas.
In conclusion, by systematically analyzing the equation y = x² - 15x + 54, we can confidently identify its corresponding graph. Understanding the quadratic form, finding the roots, and locating the vertex are crucial steps in this process. This exercise demonstrates the power of connecting algebraic equations with their visual representations, a fundamental skill in mathematical literacy.
