Parabola Y = X^2 - 6x + 5: Find Vertex & Intercepts
Let's explore the parabola defined by the equation y = x² - 6x + 5. In this article, we'll break down how to find its key features: the vertex, the y-intercept, and the x-intercepts. Understanding these elements will give you a solid grasp of the parabola's shape and position on the coordinate plane.
A) Determining the Vertex of the Parabola
When dealing with a parabola in the standard form of y = ax² + bx + c, pinpointing the vertex is crucial. The vertex, that turning point of the parabola, unveils whether the curve opens upwards or downwards and signifies its minimum or maximum value. Our given equation, y = x² - 6x + 5, falls perfectly into this form, where a = 1, b = -6, and c = 5. The x-coordinate of the vertex can be elegantly computed using the formula -b / 2a. So, let's put it into action: the x-coordinate emerges as -(-6) / (2 * 1) = 3. To unveil the y-coordinate, we simply substitute this x-value back into our original equation, yielding y = (3)² - 6(3) + 5 = 9 - 18 + 5 = -4. Hence, the vertex proudly stands at the point (3, -4). This vertex is incredibly important because it represents the minimum point of the parabola, given that the coefficient of the x² term (a) is positive (a = 1), indicating the parabola opens upwards. Understanding this foundational concept is key to analyzing quadratic functions.
To further clarify, let's delve a bit deeper into why the vertex is so significant and how its calculation connects to the broader understanding of parabolas. The formula -b / 2a for the x-coordinate of the vertex is derived from the process of completing the square, a technique used to rewrite the quadratic equation in vertex form: y = a(x - h)² + k, where (h, k) is the vertex. By completing the square, we essentially transform the equation into a form that explicitly reveals the vertex coordinates, allowing for easier analysis of the parabola's behavior. In our case, the vertex (3, -4) tells us that the lowest point on the parabola is at x = 3 and y = -4. For values of x greater or lesser than 3, the y-values will increase, creating the characteristic U-shape of the parabola. This understanding is crucial for applications such as optimization problems, where we might want to find the minimum or maximum value of a quadratic function within a certain context. For instance, if this parabola represented the profit function of a business, the vertex would tell us the point at which profit is minimized. Therefore, the vertex is not just a point; it's a key piece of information that unlocks deeper insights into the nature and applications of the parabolic function. We use this knowledge in various fields, including physics (projectile motion), engineering (designing parabolic reflectors), and economics (modeling costs and revenues). Understanding how to find and interpret the vertex is a foundational skill in mathematics.
To solidify our understanding, let's consider how the coefficients a, b, and c affect the parabola's shape and position. As we've already touched upon, the sign of a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The larger the absolute value of a, the “narrower” the parabola appears, meaning it increases (or decreases) more rapidly as we move away from the vertex. The coefficient b, on the other hand, affects the horizontal position of the parabola. A change in b will shift the vertex left or right, altering the parabola's symmetry axis. The constant term c directly corresponds to the y-intercept, as we'll discuss in the next section. It determines where the parabola intersects the y-axis. In summary, by understanding the interplay between these coefficients and the vertex formula, we gain a powerful tool for analyzing and manipulating parabolas. Imagine you needed to shift the parabola to the left or right, or make it “wider” or “narrower” – you would know exactly which coefficient to adjust and how. This level of control and understanding is a key goal in mathematical analysis. Furthermore, being able to find the vertex quickly allows you to sketch the parabola more accurately, which is important for solving problems graphically or for visualizing the behavior of the function. Remember, the vertex is not an isolated piece of information; it's a central component in the broader picture of the parabola's characteristics.
B) Identifying the Y-intercept of the Parabola
The y-intercept is where our parabola gracefully intersects the y-axis. This pivotal point occurs when x = 0. To discover it, we seamlessly substitute x = 0 into our equation y = x² - 6x + 5. This transforms our equation into y = (0)² - 6(0) + 5, simplifying beautifully to y = 5. Therefore, the y-intercept proudly stands at the point (0, 5). This intercept is significant as it gives us a direct point of reference on the graph, showing where the parabola begins its journey along the y-axis. Finding the y-intercept is often the simplest step in sketching a parabola, as it only involves substituting zero for x and evaluating the equation. This straightforward calculation provides a valuable anchor point for visualizing the graph's behavior.
Let's explore further why the y-intercept is such a straightforward calculation and its importance in the context of understanding the parabola's overall behavior. The y-intercept, occurring at x = 0, directly reveals the constant term in the quadratic equation. In our equation, y = x² - 6x + 5, the constant term is 5, which immediately tells us that the y-intercept is at y = 5. This is because when x = 0, the x² and -6x terms vanish, leaving us with just the constant term. This direct relationship makes finding the y-intercept a quick and easy task. But beyond its simplicity, the y-intercept provides valuable information about the parabola's vertical position. It tells us where the parabola crosses the vertical axis, giving us a sense of its overall height and placement on the coordinate plane. When combined with the vertex, the y-intercept helps us to sketch the parabola more accurately. For example, knowing that the vertex is at (3, -4) and the y-intercept is at (0, 5) gives us two key points that define the shape and position of the curve. The y-intercept also has practical applications in real-world scenarios. If our parabolic equation represents a physical phenomenon, such as the height of a projectile over time, the y-intercept might represent the initial height of the projectile. Therefore, the y-intercept is not just a mathematical point; it can also have a meaningful interpretation within a specific context. Understanding how to quickly identify and interpret the y-intercept is a valuable skill in both mathematical analysis and practical applications.
To broaden our perspective, consider how the y-intercept can help us compare different parabolas. If we have two parabolas with different y-intercepts, we know immediately that they cross the y-axis at different points. This can be a useful way to quickly differentiate between functions or to understand how changing the constant term in the equation affects the parabola's graph. For instance, if we were to change our equation to y = x² - 6x + 10, the y-intercept would shift upwards to (0, 10), causing the entire parabola to shift upwards as well. By understanding how the y-intercept is affected by the constant term, we gain a powerful tool for manipulating and comparing parabolic functions. Furthermore, the y-intercept can be a crucial reference point when solving problems involving parabolas. It can help us determine the range of possible solutions or to identify specific points of interest. For example, if we were trying to find the points where the parabola intersects a horizontal line, the y-intercept could give us a starting point for our analysis. In summary, the y-intercept is a simple yet powerful feature of the parabola that provides valuable information about its vertical position and overall behavior. Its ease of calculation and its direct relationship to the constant term in the equation make it an indispensable tool for analyzing quadratic functions.
C) Unveiling the X-intercepts of the Parabola
The x-intercepts are the points where the parabola intersects the x-axis, which is where y = 0. To find these crucial points, we set our equation y = x² - 6x + 5 to zero, resulting in the quadratic equation x² - 6x + 5 = 0. Now, we can employ the magic of factoring to solve for x. We seek two numbers that multiply to 5 and add up to -6. These numbers are -5 and -1. Thus, we can rewrite our equation as (x - 5)(x - 1) = 0. This equation holds true if either x - 5 = 0 or x - 1 = 0. Solving these simple equations, we find x = 5 and x = 1. Therefore, the x-intercepts are the points (1, 0) and (5, 0). These intercepts are incredibly valuable as they mark where the parabola crosses the horizontal axis, giving us a sense of its spread and roots. Finding x-intercepts helps us understand the parabola's solutions and its relation to the x-axis. This knowledge is critical in various mathematical and real-world applications.
Let's delve deeper into the significance of x-intercepts and explore the different methods we can use to find them. While factoring is an efficient method when applicable, not all quadratic equations can be easily factored. In such cases, we can turn to the quadratic formula, a powerful tool that works for any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is given by: x = [-b ± √(b² - 4ac)] / 2a. Applying this formula to our equation x² - 6x + 5 = 0, we have a = 1, b = -6, and c = 5. Substituting these values into the formula, we get: x = [6 ± √((-6)² - 4 * 1 * 5)] / (2 * 1), which simplifies to x = [6 ± √(36 - 20)] / 2 = [6 ± √16] / 2 = [6 ± 4] / 2. This gives us two solutions: x = (6 + 4) / 2 = 5 and x = (6 - 4) / 2 = 1, confirming our previous results from factoring. The quadratic formula highlights an important aspect of x-intercepts: a parabola can have two distinct x-intercepts, one x-intercept (when the discriminant, b² - 4ac, is zero), or no x-intercepts (when the discriminant is negative). This is because the discriminant determines the nature of the roots of the quadratic equation. Understanding the discriminant is crucial for predicting the number and type of x-intercepts a parabola will have. Furthermore, the x-intercepts have practical interpretations in various contexts. For example, if the parabola represents the trajectory of a projectile, the x-intercepts might represent the points where the projectile lands on the ground. Therefore, finding x-intercepts is not just a mathematical exercise; it can provide valuable insights into real-world phenomena.
To expand our understanding, let's consider how the x-intercepts relate to the vertex and the axis of symmetry of the parabola. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The x-intercepts, if they exist, are equidistant from the axis of symmetry. In our case, the vertex is at (3, -4), so the axis of symmetry is the line x = 3. The x-intercepts are at x = 1 and x = 5, which are both 2 units away from x = 3. This symmetry is a fundamental property of parabolas and can be used to check the accuracy of our calculations. If we found x-intercepts that were not symmetrical about the axis of symmetry, we would know that we had made an error. The relationship between the x-intercepts, the vertex, and the axis of symmetry also provides a visual way to understand the parabola's behavior. By knowing the vertex and the x-intercepts, we can accurately sketch the parabola's shape and position on the coordinate plane. In summary, the x-intercepts are key features of the parabola that provide valuable information about its roots, its relationship to the x-axis, and its symmetry. By mastering the techniques for finding x-intercepts and understanding their significance, we gain a deeper appreciation for the nature and applications of quadratic functions.
Conclusion
In conclusion, by systematically finding the vertex, y-intercept, and x-intercepts of the parabola y = x² - 6x + 5, we've gained a comprehensive understanding of its shape and position. The vertex (3, -4) pinpoints the parabola's minimum point, the y-intercept (0, 5) marks its intersection with the y-axis, and the x-intercepts (1, 0) and (5, 0) reveal where it crosses the x-axis. These key features enable us to accurately sketch and analyze the parabola's behavior, a crucial skill in mathematics and various real-world applications. For further exploration of parabolas and quadratic functions, consider visiting Khan Academy's Algebra resources.
