How To Find The Vertex, Zeros, And Y-Intercept Of Y=-x²+4x+32
Find the vertex, zeros, and y-intercept of the graph of the quadratic equation y=-x²+4x+32.
This article will guide you through the process of finding the vertex, zeros (also known as x-intercepts), and y-intercept of the quadratic function y = -x² + 4x + 32. Understanding these key features allows us to accurately graph the parabola represented by this equation and analyze its behavior. We'll cover the necessary formulas and techniques in a step-by-step manner, making it easy to follow along. So, whether you're a student learning about quadratic functions or just looking to brush up on your math skills, this guide will provide you with the tools you need to succeed. Let's dive in and explore the fascinating world of parabolas!
Understanding Quadratic Functions
Before we jump into the specifics of finding the vertex, zeros, and y-intercept, let's briefly discuss quadratic functions in general. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually x) is 2. The standard form of a quadratic function is y = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, a U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. In our case, the function y = -x² + 4x + 32 has a = -1, b = 4, and c = 32. Since a is negative, we know the parabola opens downwards. Understanding this basic structure is the first step in deciphering the characteristics of our specific function.
Key features of a quadratic function's graph include the vertex, which is the highest or lowest point on the parabola (depending on whether it opens upwards or downwards), the zeros (or x-intercepts), which are the points where the parabola intersects the x-axis, and the y-intercept, which is the point where the parabola intersects the y-axis. Finding these features helps us understand the parabola's position and shape in the coordinate plane. Each of these points provides valuable information about the function's behavior and real-world applications it might model. For instance, the vertex can represent the maximum height of a projectile, and the zeros can represent the points where a profit function breaks even. The y-intercept, on the other hand, can indicate the initial value of a quantity being modeled by the quadratic function. Therefore, mastering the techniques for finding these key features is crucial for anyone working with quadratic functions.
Finding the Vertex
The vertex is a crucial point on the parabola, representing either the maximum or minimum value of the quadratic function. For a parabola that opens downwards (like ours), the vertex is the highest point. The vertex of a parabola in the form y = ax² + bx + c can be found using the vertex formula. The x-coordinate of the vertex, denoted as h, is given by the formula h = -b / 2a. Once we find h, we can substitute it back into the original equation to find the y-coordinate of the vertex, denoted as k. Thus, the vertex is the point (h, k). In our example, y = -x² + 4x + 32, we have a = -1 and b = 4. Plugging these values into the formula for h, we get h = -4 / (2 * -1) = -4 / -2 = 2. This tells us that the x-coordinate of our vertex is 2. Now we need to find the corresponding y-coordinate. Let's substitute x = 2 into our original equation.
To find the y-coordinate (k) of the vertex, we substitute x = 2 into the equation y = -x² + 4x + 32: k = -(2)² + 4(2) + 32. Simplifying this expression, we get k = -4 + 8 + 32 = 36. Therefore, the y-coordinate of the vertex is 36. Combining the x-coordinate (2) and the y-coordinate (36), we find that the vertex of the parabola is (2, 36). This means the highest point on the parabola represented by the function y = -x² + 4x + 32 is at the point (2, 36). Knowing the vertex is essential for sketching the parabola and understanding its behavior. It gives us a central point around which the parabola is symmetrical, and it also indicates the maximum value the function reaches. In practical applications, the vertex might represent the peak height of a projectile or the maximum profit in a business scenario. The vertex is a critical piece of information for analyzing and interpreting quadratic functions and their applications.
Determining the Zeros (X-Intercepts)
Zeros, also known as x-intercepts, are the points where the parabola intersects the x-axis. At these points, the y-value is zero. To find the zeros of the quadratic function y = -x² + 4x + 32, we need to solve the equation -x² + 4x + 32 = 0 for x. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. In this case, factoring is a straightforward approach. We need to find two numbers that multiply to -32 and add up to 4. These numbers are 8 and -4. So, we can rewrite the equation as -(x² - 4x - 32) = 0. Factoring the quadratic expression inside the parentheses, we get -(x - 8)(x + 4) = 0. This equation is satisfied when either (x - 8) = 0 or (x + 4) = 0.
Setting each factor equal to zero, we have two equations to solve: x - 8 = 0 and x + 4 = 0. Solving x - 8 = 0 gives us x = 8, and solving x + 4 = 0 gives us x = -4. Therefore, the zeros of the function y = -x² + 4x + 32 are x = 8 and x = -4. These zeros correspond to the points where the parabola crosses the x-axis, which are (8, 0) and (-4, 0). The zeros provide essential information about the quadratic function's behavior. They indicate where the function's value is zero, which can be significant in various applications. For instance, if the function represents the profit of a business, the zeros represent the break-even points. If the function represents the height of a projectile, the zeros might represent the points where the projectile hits the ground. Understanding how to find and interpret the zeros of a quadratic function is crucial for its practical application and analysis.
Identifying the Y-Intercept
The y-intercept is the point where the parabola intersects the y-axis. At this point, the x-value is zero. To find the y-intercept of the quadratic function y = -x² + 4x + 32, we simply need to substitute x = 0 into the equation and solve for y. This is because any point on the y-axis has an x-coordinate of 0. So, substituting x = 0 into our equation, we get y = -(0)² + 4(0) + 32. Simplifying this expression, we get y = 0 + 0 + 32 = 32. Therefore, the y-intercept of the function y = -x² + 4x + 32 is 32. This means the parabola intersects the y-axis at the point (0, 32).
The y-intercept provides valuable information about the quadratic function. It represents the value of the function when x is zero, which can have various interpretations depending on the context. For example, if the function represents the population of a species over time, the y-intercept would represent the initial population. If the function represents the cost of production, the y-intercept might represent the fixed costs. The y-intercept, in conjunction with the vertex and zeros, gives us a comprehensive picture of the parabola's behavior and its position in the coordinate plane. Knowing the y-intercept allows us to accurately sketch the parabola and interpret its significance in various real-world scenarios. It's a crucial piece of information for understanding and analyzing quadratic functions and their applications.
Conclusion
In summary, we've successfully found the vertex, zeros, and y-intercept of the quadratic function y = -x² + 4x + 32. The vertex is (2, 36), representing the maximum point of the parabola. The zeros are (-4, 0) and (8, 0), indicating where the parabola intersects the x-axis. The y-intercept is (0, 32), showing where the parabola intersects the y-axis. These key features provide a complete picture of the parabola's shape, position, and behavior in the coordinate plane. Understanding how to find these features is crucial for analyzing and applying quadratic functions in various fields, from physics to economics.
By mastering these techniques, you can confidently tackle quadratic functions and their applications. The vertex gives you the maximum or minimum value of the function, the zeros indicate the points where the function's value is zero, and the y-intercept provides the function's value when x is zero. These pieces of information, when combined, allow you to sketch the graph of the parabola and understand its real-world implications. Whether you're solving a mathematical problem, modeling a physical phenomenon, or analyzing economic data, the ability to find the vertex, zeros, and y-intercept of a quadratic function is an invaluable skill. Remember to practice these steps and apply them to different quadratic functions to solidify your understanding and build confidence in your problem-solving abilities.