Finding The Y-Intercept Of F(x) = -√(5x + 2) - 1 A Step-by-Step Guide

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In the realm of mathematics, the y-intercept holds significant importance as a crucial point on a graph. It represents the spot where a function intersects the y-axis, providing valuable insights into the function's behavior and characteristics. For the function f(x) = -√(5x + 2) - 1, determining the y-intercept involves a straightforward yet insightful process. This article delves into the step-by-step method of finding the y-intercept, elucidating the underlying concepts and providing a comprehensive understanding of the solution. We will explore the significance of the y-intercept in the context of this specific function, and how it contributes to a broader understanding of mathematical functions and their graphical representations.

Defining the Y-Intercept

At its core, the y-intercept is the point where a graph crosses the y-axis. This intersection occurs when the x-coordinate is zero. In other words, to find the y-intercept of a function, we simply need to substitute x = 0 into the function's equation and solve for y. The resulting y-value will be the y-coordinate of the y-intercept. This concept is fundamental to understanding the behavior of functions and their graphical representations. The y-intercept is not merely a point on the graph; it's a key characteristic that helps us visualize the function's trajectory and its relationship with the coordinate axes. It's often the first point we identify when sketching a graph, as it provides a solid anchor for the curve or line. Furthermore, the y-intercept can reveal important information about the function's initial value or starting point, particularly in real-world applications where functions are used to model various phenomena.

Step-by-Step Solution

To find the y-intercept of the function f(x) = -√(5x + 2) - 1, we follow these steps:

  1. Substitute x = 0 into the function: This is the fundamental step in finding the y-intercept. By setting x = 0, we are essentially looking for the point on the function's graph where it intersects the y-axis. This substitution transforms the function's equation into an expression that we can solve for the corresponding y-value. It's a direct application of the definition of the y-intercept, ensuring that we are focusing on the specific point of intersection with the vertical axis.

    • f(0) = -√(5(0) + 2) - 1

  2. Simplify the expression: Now, we simplify the expression obtained after substituting x = 0. This involves performing the arithmetic operations within the square root and then evaluating the entire expression. The simplification process ensures that we arrive at the most concise and accurate representation of the y-value. It's a crucial step in isolating the y-value and determining the precise coordinates of the y-intercept. Simplifying the expression not only provides the numerical value of the y-intercept but also enhances our understanding of the function's behavior at that specific point.

    • f(0) = -√(2) - 1

  3. The y-intercept is (0, -√2 - 1): The result, f(0) = -√2 - 1, gives us the y-coordinate of the y-intercept. Since we substituted x = 0, the x-coordinate is 0. Therefore, the y-intercept is the point (0, -√2 - 1). This point represents the exact location where the function's graph crosses the y-axis. It's a critical piece of information for visualizing the function's behavior and understanding its relationship with the coordinate system. The y-intercept, along with other key points such as the x-intercept and vertex, provides a comprehensive picture of the function's graph and its characteristics.

Significance of the Y-Intercept

The y-intercept is not just a point on a graph; it carries significant meaning and provides valuable information about the function. It represents the value of the function when the input (x) is zero. In many real-world applications, this translates to the initial value or starting point of a process or phenomenon modeled by the function. For example, if the function represents the population of a species over time, the y-intercept would represent the initial population at time zero. Similarly, if the function models the height of a projectile, the y-intercept would represent the initial height from which it was launched. Understanding the y-intercept is crucial for interpreting the function's behavior and its practical implications. It allows us to make predictions about the function's output for different input values and to gain insights into the underlying process or system it represents. The y-intercept, therefore, serves as a fundamental anchor point for analyzing and understanding mathematical models.

Analyzing the Given Options

Given the options:

  • A. none
  • B. (0, -√2 - 1)
  • C. (0, √2 - 1)
  • D. (-√2 - 1, 0)

Based on our step-by-step solution, the correct y-intercept is (0, -√2 - 1), which corresponds to option B. Options A, C, and D are incorrect because they either do not represent a valid point or do not satisfy the condition of being the y-intercept (i.e., having an x-coordinate of 0). Option D, in particular, represents an x-intercept, which is the point where the graph crosses the x-axis, not the y-axis. The ability to distinguish between the y-intercept and other key points on a graph, such as the x-intercept, is essential for accurate interpretation and analysis of functions. Understanding the definition and significance of each point allows us to build a comprehensive understanding of the function's behavior and its graphical representation.

Conclusion

In conclusion, finding the y-intercept of a function is a fundamental skill in mathematics. For the function f(x) = -√(5x + 2) - 1, the y-intercept is (0, -√2 - 1). This point provides valuable information about the function's behavior and its intersection with the y-axis. By understanding the concept of the y-intercept and the steps involved in finding it, we can gain a deeper understanding of mathematical functions and their graphical representations. The y-intercept serves as a crucial anchor point for visualizing and analyzing functions, providing insights into their initial values and overall behavior. Mastering the techniques for finding the y-intercept is essential for success in various mathematical and scientific disciplines, as it enables us to interpret and apply mathematical models to real-world phenomena effectively.