Finding Parabola Equation Zeros X=-2 X=4 Y-intercept (0,-16)

In mathematics, a parabola is a U-shaped curve that is the graph of a quadratic function. Quadratic functions are of the form f(x) = ax² + bx + c, where a, b, and c are constants and a is not equal to zero. Understanding parabolas is fundamental in various areas of mathematics and its applications, including physics, engineering, and computer graphics. A key aspect of working with parabolas involves determining the equation of a parabola given specific characteristics, such as its zeros (x-intercepts) and y-intercept. This article delves into the process of finding the equation of a parabola when provided with its zeros and y-intercept, with a focus on a specific example. We will explore the underlying concepts, step-by-step methods, and practical considerations to solve this type of problem effectively.

Before diving into the solution, it's important to define the zeros and y-intercept of a function. Zeros, also known as roots or x-intercepts, are the points where the parabola intersects the x-axis. At these points, the function's value is zero, meaning f(x) = 0. Knowing the zeros of a parabola provides crucial information about its position and orientation in the coordinate plane. In contrast, the y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0, and the y-intercept is the value of the function at this point, f(0). The y-intercept helps to further define the parabola's location and its overall shape. Together, the zeros and y-intercept provide significant clues for determining the unique equation of the parabola.

The general form of a quadratic equation is f(x) = ax² + bx + c, where a, b, and c are constants that determine the shape and position of the parabola. The coefficient a dictates the parabola's direction (whether it opens upwards or downwards) and its vertical stretch or compression. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The constants b and c influence the parabola's horizontal and vertical position, respectively. The c term specifically represents the y-intercept of the parabola because when x = 0, f(0) = c. While the general form is useful, another form, the factored form, is particularly helpful when zeros are known. The factored form of a quadratic equation is f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the zeros of the parabola. This form directly incorporates the zeros, making it easier to construct the equation when the zeros are given. By understanding both the general and factored forms, we can effectively manipulate and solve quadratic equations based on the information provided.

Consider the problem of finding the specific function that represents a parabola with zeros at x = -2 and x = 4, and a y-intercept at (0, -16). This problem is a classic example of how to use key features of a parabola—namely, its zeros and y-intercept—to identify its equation. To solve this, we need to find the quadratic function f(x) = ax² + bx + c that satisfies these conditions. The zeros provide the x-values where the parabola intersects the x-axis, and the y-intercept gives the point where the parabola intersects the y-axis. Using this information, we can set up a system of equations or use the factored form of a quadratic equation to determine the coefficients a, b, and c. The challenge lies in correctly applying the given information to form the equation and then verifying that the resulting function indeed matches all the provided conditions. This exercise is not only a fundamental skill in algebra but also a crucial concept in understanding and modeling real-world phenomena using quadratic functions.

The factored form of a quadratic equation is an invaluable tool when dealing with problems involving zeros. Given the zeros x = -2 and x = 4, we can express the parabola in the form f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the zeros. Substituting the given zeros, we get f(x) = a(x - (-2))(x - 4), which simplifies to f(x) = a(x + 2)(x - 4). This form directly incorporates the x-intercepts into the equation, leaving only the coefficient a to be determined. The coefficient a affects the vertical stretch or compression of the parabola and whether it opens upwards or downwards. To find the value of a, we utilize the given y-intercept. The y-intercept is the point where the parabola crosses the y-axis, which occurs when x = 0. Thus, the y-intercept (0, -16) means that f(0) = -16. By substituting these values into the factored form, we can solve for a. This method is efficient because it directly uses the zeros to build the equation, making it a straightforward process to find the parabola's function.

To determine the value of a in the factored form f(x) = a(x + 2)(x - 4), we use the given y-intercept (0, -16). This point tells us that when x = 0, f(x) = -16. Substituting these values into the equation, we get -16 = a(0 + 2)(0 - 4). Simplifying this, we have -16 = a(2)(-4), which further simplifies to -16 = -8a. To solve for a, we divide both sides of the equation by -8, resulting in a = 2. Now that we have found the value of a, we can substitute it back into the factored form to obtain the complete equation of the parabola. This step is crucial as it finalizes the equation, ensuring that it fits all the given conditions—both the zeros and the y-intercept. With a determined, we can confidently express the parabola's function and move on to verifying and simplifying the equation.

Now that we have a = 2, we substitute this value back into the factored form f(x) = a(x + 2)(x - 4), resulting in f(x) = 2(x + 2)(x - 4). To obtain the standard quadratic form f(x) = ax² + bx + c, we need to expand and simplify this expression. First, we multiply the binomials (x + 2) and (x - 4): (x + 2)(x - 4) = x² - 4x + 2x - 8. Combining like terms, we get x² - 2x - 8. Next, we multiply the entire expression by 2: 2(x² - 2x - 8) = 2x² - 4x - 16. Thus, the simplified form of the quadratic function is f(x) = 2x² - 4x - 16. This form is useful for identifying the coefficients a, b, and c, and for further analysis of the parabola's properties. The expanded form also allows for easy comparison with given options or other forms of the equation, ensuring that the solution is correct and matches the problem's requirements.

After finding the equation f(x) = 2x² - 4x - 16, it is essential to verify that this function indeed satisfies the given conditions: zeros at x = -2 and x = 4, and a y-intercept at (0, -16). To verify the zeros, we substitute x = -2 and x = 4 into the function and check if f(x) = 0. For x = -2, we have f(-2) = 2(-2)² - 4(-2) - 16 = 2(4) + 8 - 16 = 8 + 8 - 16 = 0. For x = 4, we have f(4) = 2(4)² - 4(4) - 16 = 2(16) - 16 - 16 = 32 - 16 - 16 = 0. Both zeros are verified. To verify the y-intercept, we substitute x = 0 into the function: f(0) = 2(0)² - 4(0) - 16 = 0 - 0 - 16 = -16. The y-intercept (0, -16) is also verified. Since the function satisfies all the given conditions, we can confidently conclude that f(x) = 2x² - 4x - 16 is the correct equation for the parabola. This verification step is crucial for ensuring accuracy and completeness in solving the problem.

In conclusion, finding the equation of a parabola given its zeros and y-intercept involves a systematic approach that utilizes the properties of quadratic functions. Starting with the factored form f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the zeros, we incorporate the given zeros into the equation. Then, using the y-intercept (0, f(0)), we substitute x = 0 and f(0) into the equation to solve for the coefficient a. Once a is determined, we can expand and simplify the equation to the standard quadratic form f(x) = ax² + bx + c. Finally, we verify the solution by substituting the zeros and the x-coordinate of the y-intercept back into the equation to ensure they satisfy the given conditions. This process not only helps in solving specific problems but also enhances understanding of quadratic functions and their graphical representation as parabolas. By mastering these techniques, one can effectively tackle a wide range of problems involving parabolas and their applications in various fields.

The correct function that represents the parabola with zeros at x = -2 and x = 4, and a y-intercept at (0, -16) is f(x) = 2x² - 4x - 16.