Finding The Parabola Equation With A=1, B=3, And C=6

In the realm of mathematics, understanding the parabola equation is crucial. A parabola is a symmetrical U-shaped curve, and its equation is typically represented in the standard form: ax² + bx + c = 0. The coefficients a, b, and c determine the shape and position of the parabola on the coordinate plane. In this article, we will delve into the process of identifying the correct equation for a parabola given specific values for a, b, and c. Specifically, we aim to find the equation that fits a parabola with an a-value of 1, a b-value of 3, and a c-value of 6. Understanding the role of each coefficient is essential in accurately representing parabolic functions and their graphical representations. The coefficient a dictates the parabola's direction (whether it opens upwards or downwards) and its width. A positive a indicates an upward-opening parabola, while a negative a indicates a downward-opening one. The magnitude of a affects the parabola's width; a larger magnitude results in a narrower parabola, and a smaller magnitude results in a wider one. The coefficient b influences the parabola's axis of symmetry, which is a vertical line that divides the parabola into two mirror-image halves. The b coefficient, along with a, determines the x-coordinate of the vertex, which is the parabola's highest or lowest point. The coefficient c represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. It is the value of y when x equals 0. This constant term plays a vital role in positioning the parabola vertically on the coordinate plane. By carefully considering these roles, we can accurately identify the parabola equation that fits the given coefficients. We will explore the provided options and systematically eliminate those that do not match the specified conditions. This approach will enable us to pinpoint the correct equation and deepen our understanding of parabolic functions and their characteristics.

Understanding the Standard Form of a Parabola Equation

The standard form of a parabola equation is expressed as ax² + bx + c = 0. This form is fundamental in understanding the properties and behavior of parabolic functions. Each coefficient in this equation plays a significant role in defining the parabola's characteristics. The leading coefficient, a, is particularly important as it determines the direction and width of the parabola. When a is positive, the parabola opens upwards, resembling a U-shape. Conversely, when a is negative, the parabola opens downwards, resembling an inverted U-shape. The magnitude of a also influences the parabola's width; a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider parabola. This relationship between a and the parabola's shape is crucial in visualizing and interpreting parabolic functions. The coefficient b affects the parabola's horizontal position and is instrumental in determining the axis of symmetry. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. The x-coordinate of the vertex, which is the point where the parabola changes direction, is given by -b/(2a). This formula highlights the interplay between a and b in positioning the parabola along the x-axis. Understanding the influence of b is essential for accurately sketching and analyzing parabolic graphs. The constant term, c, represents the y-intercept of the parabola. The y-intercept is the point where the parabola intersects the y-axis, and it occurs when x is equal to 0. In the standard form equation, setting x to 0 directly reveals that y equals c. This simple relationship makes c a straightforward indicator of the parabola's vertical position. By grasping the individual roles of a, b, and c, one can effectively analyze and manipulate parabolic equations to fit specific conditions and requirements. This understanding forms the basis for solving a variety of problems involving parabolas, from finding roots and vertices to modeling real-world phenomena such as projectile motion and reflector designs. The standard form of a parabola equation provides a clear framework for these analyses, making it an indispensable tool in mathematics and its applications.

Identifying the Correct Equation with Given Coefficients

To identify the correct parabola equation that fits the given coefficients a = 1, b = 3, and c = 6, we must substitute these values into the standard form equation, ax² + bx + c = 0. This process involves replacing the variables a, b, and c with their respective numerical values, which in this case are 1, 3, and 6. The resulting equation will represent a specific parabola defined by these coefficients. Substituting a = 1, b = 3, and c = 6 into the standard form equation yields: 1x² + 3x + 6 = 0, which simplifies to x² + 3x + 6 = 0. This equation is the unique representation of a parabola with the specified coefficients. It is essential to compare this derived equation with the provided options to determine which one matches. The correct option will have the same coefficients for the x², x, and constant terms as our derived equation. Any deviation in these coefficients would indicate that the option does not represent the parabola we are seeking. By systematically comparing the derived equation with the options, we can accurately identify the one that fits the given conditions. This process not only helps in finding the correct equation but also reinforces the understanding of how different coefficients affect the parabola's shape and position. In this specific scenario, the correct equation, x² + 3x + 6 = 0, embodies a parabola that opens upwards (since a is positive) and has a specific curvature and vertical placement determined by the values of b and c. Recognizing the significance of each coefficient in the equation is crucial for accurately representing and interpreting parabolic functions. The methodical substitution and comparison process ensures that the identified equation truly reflects the intended parabolic characteristics. This skill is invaluable in various mathematical and real-world applications where parabolas are used to model phenomena such as projectile trajectories, bridge arches, and satellite dish designs.

Analyzing the Provided Options

Now, let's analyze the provided options to determine which equation matches the parabola equation we derived, x² + 3x + 6 = 0. Each option presents a different combination of terms, and we need to carefully examine their coefficients to see if they align with our target equation. This process involves comparing the coefficients of the x², x, and constant terms in each option with those in our derived equation. The first option is 0 = 2x - 5 + x⁴. This equation is a quartic equation due to the x⁴ term, not a quadratic equation that represents a parabola. Therefore, this option can be immediately eliminated. It does not fit the standard form of a parabola equation, which requires a maximum power of 2 for the variable x. The presence of x⁴ indicates a fundamentally different type of curve. The second option is 0 = x - 3 - 5x². Rearranging this equation into the standard form, we get -5x² + x - 3 = 0. Here, the coefficient of x² is -5, which does not match our required a-value of 1. Additionally, the coefficient of x is 1, and the constant term is -3, which do not match our b-value of 3 and c-value of 6, respectively. Thus, this option is also incorrect. The negative a value would indicate a downward-opening parabola, which is not what we are looking for. The third option is 0 = 3x - 5 - x². Rearranging this equation into the standard form, we get -x² + 3x - 5 = 0. The coefficient of x² is -1, which again does not match our required a-value of 1. The coefficients of x and the constant term also do not align with our target equation. This option can be eliminated as well. The negative a value here, like in the previous option, signifies a downward-opening parabola, which is inconsistent with our desired conditions. The fourth option is 0 = 3x + 5x². Rearranging this equation into the standard form, we get 5x² + 3x + 0 = 0. The coefficient of x² is 5, which does not match our required a-value of 1. The coefficient of x matches our b-value of 3, but the constant term is 0, which does not match our c-value of 6. Therefore, this option is incorrect. By systematically analyzing each option and comparing its coefficients with our derived equation, we can confidently eliminate the incorrect ones. This process highlights the importance of accurately identifying the coefficients and their roles in defining the parabola's shape and position. The correct equation must have the exact same coefficients for the x², x, and constant terms as our target equation. This meticulous approach ensures that we select the equation that truly represents the parabola with the given characteristics.

Conclusion: The Correct Parabola Equation

In conclusion, after a thorough analysis of the provided options, we can confidently state that none of the given equations fit a parabola with an a-value of 1, a b-value of 3, and a c-value of 6. Our derived parabola equation, based on the standard form ax² + bx + c = 0, is x² + 3x + 6 = 0. This equation uniquely represents a parabola with the specified coefficients. The discrepancy arises because none of the provided options match this exact equation. Each option either has a different coefficient for the x² term, the x term, or the constant term, or a combination thereof. This exercise underscores the importance of precisely matching coefficients when identifying the equation of a parabola. The coefficients a, b, and c play distinct roles in shaping the parabola's characteristics, including its direction, width, and position on the coordinate plane. Any deviation in these coefficients results in a different parabola. Understanding the standard form of the parabola equation and the significance of each coefficient is crucial for accurately representing and interpreting parabolic functions. This knowledge enables us to solve various mathematical problems involving parabolas, such as finding their roots, vertices, and axes of symmetry. It also facilitates the application of parabolas in real-world scenarios, such as modeling projectile motion, designing reflective surfaces, and analyzing structural arches. The process of systematically comparing equations and their coefficients is a valuable skill in mathematics. It reinforces the importance of attention to detail and accuracy in problem-solving. In this case, the methodical approach allowed us to eliminate incorrect options and confidently conclude that none of them matched the target parabola equation. This conclusion not only answers the specific question but also reinforces the broader understanding of parabolic functions and their mathematical representation. The ability to accurately identify and manipulate parabola equations is a fundamental skill in algebra and calculus, with far-reaching applications in various scientific and engineering disciplines.