Finding The Y-Intercept Of Y=6(x-1/2)(x+3) In The XY-Plane

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In the realm of coordinate geometry, identifying key features of equations, such as intercepts, is a fundamental skill. Let's delve into a specific problem involving a quadratic equation and its graphical representation on the xy-plane. Our goal is to determine the y-intercept of the graph defined by the equation y = 6(x - 1/2)(x + 3). The y-intercept, a crucial point, marks where the graph intersects the y-axis. Understanding how to find this point is essential for analyzing and interpreting quadratic functions.

Understanding the Y-Intercept

The y-intercept is the point where a graph intersects the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept of any equation, we substitute x = 0 into the equation and solve for y. This process is rooted in the fundamental principles of coordinate geometry, where each point on the plane is defined by its x and y coordinates. The y-intercept provides valuable information about the function's behavior, particularly its initial value or starting point on the y-axis. For quadratic equations, the y-intercept, along with the x-intercepts (roots) and the vertex, helps in sketching the parabola accurately.

Step-by-Step Solution

To find the y-intercept of the given equation, y = 6(x - 1/2)(x + 3), we will follow these steps:

  1. Substitute x = 0 into the equation: This is the foundational step, replacing the variable x with the numerical value zero. This substitution leverages the defining characteristic of the y-intercept, where the x-coordinate is inherently zero. By making this substitution, we effectively isolate the y-value at the point where the graph intersects the y-axis.

  2. Simplify the equation: After substituting, the equation becomes y = 6(0 - 1/2)(0 + 3). This simplification involves basic arithmetic operations, including subtraction and addition within the parentheses. The order of operations (PEMDAS/BODMAS) dictates that we address the expressions within the parentheses before proceeding with multiplication. This step transforms the equation into a more manageable form, paving the way for the subsequent calculation of the y-value.

  3. Perform the arithmetic operations: We further simplify the equation: y = 6(-1/2)(3). This step involves multiplying the numerical values obtained after the initial simplification. Multiplication is a fundamental arithmetic operation, and its accurate execution is crucial for arriving at the correct y-intercept. The negative sign associated with the -1/2 term is carefully considered during the multiplication process, ensuring that the sign of the final result is correct.

  4. Calculate the final value of y: Multiplying the numbers gives us y = 6 * (-1.5) = -9. This final calculation yields the y-coordinate of the y-intercept. The arithmetic operation involves multiplying a whole number (6) by a negative fraction (-1.5). The result is a negative value, which indicates that the y-intercept lies below the x-axis on the coordinate plane. This calculated y-value represents the vertical coordinate where the parabola intersects the y-axis.

Therefore, the y-intercept is (0, -9). This point represents the precise location where the parabola defined by the equation y = 6(x - 1/2)(x + 3) crosses the y-axis. The y-coordinate, -9, indicates that the point of intersection is 9 units below the x-axis. This y-intercept provides a valuable reference point for sketching the graph of the quadratic function and understanding its behavior.

Detailed Calculation

Let's break down the calculation step by step:

  • Step 1: Substitute x = 0

    y = 6(x - 1/2)(x + 3)

    y = 6(0 - 1/2)(0 + 3)

    This substitution is the cornerstone of finding the y-intercept, transforming the equation into a form where only the y-value remains unknown. It directly applies the definition of the y-intercept as the point where the graph crosses the y-axis, which occurs when the x-coordinate is zero. This step sets the stage for the subsequent algebraic manipulation and calculation.

  • Step 2: Simplify inside the parentheses

    y = 6(-1/2)(3)

    This simplification involves performing the arithmetic operations within the parentheses. The expression (0 - 1/2) simplifies to -1/2, and the expression (0 + 3) simplifies to 3. This step streamlines the equation by reducing the complexity of the expressions within the parentheses, making the subsequent multiplication process more straightforward. The negative sign associated with the -1/2 term is carefully maintained, as it will play a crucial role in determining the sign of the final result.

  • Step 3: Multiply the numbers

    y = 6 * (-1/2) * 3

    y = -3 * 3

    y = -9

    This step involves the multiplication of the numerical values. First, we multiply 6 by -1/2, which yields -3. Then, we multiply -3 by 3, which results in -9. This step accurately executes the multiplication operations, adhering to the rules of arithmetic. The negative sign is carefully tracked throughout the multiplication process, ensuring that the final result has the correct sign. The result, y = -9, represents the y-coordinate of the y-intercept.

Thus, the y-intercept is (0, -9).

Conclusion

In conclusion, the y-intercept of the graph of the equation y = 6(x - 1/2)(x + 3) is (0, -9). This point is crucial for understanding the behavior of the quadratic function and accurately sketching its graph. By substituting x = 0 into the equation and solving for y, we can easily determine the y-intercept. This process demonstrates a fundamental principle in coordinate geometry, highlighting the relationship between algebraic equations and their graphical representations. The y-intercept, along with other key features such as the x-intercepts and the vertex, provides a comprehensive understanding of the quadratic function's characteristics. Understanding how to find the y-intercept of a graph is a crucial skill in algebra and calculus, applicable to various mathematical and real-world problems. This skill allows for the analysis and interpretation of functions, providing valuable insights into their behavior and properties. The y-intercept, as a specific point on the graph, serves as a reference for understanding the function's initial value or starting point, particularly in contexts where the function models a real-world phenomenon. Moreover, the concept of the y-intercept extends beyond quadratic functions and is applicable to a wide range of mathematical functions, making it a fundamental concept in mathematical analysis. By mastering the technique of finding the y-intercept, students can enhance their problem-solving abilities and gain a deeper appreciation for the connection between algebra and geometry. The application of the y-intercept can be seen in many areas. For example, in physics, it can represent the initial position of an object. In economics, it might represent the fixed costs of a business. Understanding the y-intercept, therefore, is not just a mathematical exercise, but a tool that can be used to interpret and understand the world around us.