Finding The Y-intercept Of Y=-x^2+8x-4 A Comprehensive Guide

In the realm of mathematics, quadratic functions hold a prominent position, their graphs gracefully tracing parabolic curves. Understanding the key features of these parabolas, such as their intercepts, is crucial for comprehending the function's behavior and its relationship to the coordinate plane. Among these intercepts, the y-intercept stands out as a particularly insightful point, revealing where the parabola intersects the vertical axis. In this comprehensive exploration, we will delve into the intricacies of determining the y-intercept of a quadratic function, using the specific example of $y = -x^2 + 8x - 4$ as our guiding star.

To embark on this mathematical journey, let's first define what a y-intercept truly represents. In essence, the y-intercept is the point where a graph intersects the y-axis. This intersection occurs when the $x$-coordinate is equal to zero. Therefore, to find the y-intercept of any function, we simply substitute $x = 0$ into the function's equation and solve for $y$. This substitution effectively isolates the $y$-value at the point where the graph crosses the y-axis.

Now, let's apply this principle to our quadratic function, $y = -x^2 + 8x - 4$. To find its y-intercept, we will set $x = 0$ and solve for $y$: $y = -(0)^2 + 8(0) - 4$. Simplifying this equation, we get $y = 0 + 0 - 4$, which further reduces to $y = -4$. This elegant calculation reveals that the y-intercept of the parabola is located at the point $(0, -4)$. This point signifies the exact location where the parabolic curve intersects the y-axis, providing us with a crucial anchor point for visualizing and understanding the function's graph.

To solidify our understanding, let's meticulously walk through the steps involved in finding the y-intercept of the quadratic function $y = -x^2 + 8x - 4$. This step-by-step approach will not only reinforce the concept but also provide a clear and repeatable method for tackling similar problems in the future. Each step is designed to be clear, concise, and easily understandable, ensuring that you grasp the underlying logic and mathematical principles.

Step 1: Understand the Concept of the $y$-intercept

The cornerstone of our approach lies in a clear understanding of the y-intercept. As we've established, the y-intercept is the point where the graph of a function intersects the y-axis. This intersection occurs precisely when the $x$-coordinate is zero. Visualizing this concept is key. Imagine the y-axis as a vertical line slicing through the coordinate plane. The point where our parabola crosses this line is the y-intercept we seek. Recognizing this fundamental relationship between the y-intercept and the y-axis sets the stage for our subsequent calculations.

Step 2: Substitute $x = 0$ into the Equation

The next crucial step is the substitution. We replace every instance of $x$ in the equation $y = -x^2 + 8x - 4$ with the value zero. This substitution is the mathematical embodiment of our understanding that the y-intercept occurs when $x = 0$. The equation now transforms into $y = -(0)^2 + 8(0) - 4$. This substitution effectively isolates the y-value that corresponds to the y-intercept. It's a simple yet powerful step that unlocks the solution.

Step 3: Simplify the Equation

With the substitution complete, we now embark on the simplification process. This involves performing the arithmetic operations to isolate $y$. We begin by evaluating the terms involving zero. $-(0)^2$ is simply zero, and $8(0)$ is also zero. Our equation now becomes $y = 0 + 0 - 4$. This simplification process reduces the equation to its essence, making the final calculation straightforward. The goal is to systematically eliminate terms until we arrive at a single value for $y$.

Step 4: Solve for $y$

The final step is the culmination of our efforts. We perform the remaining arithmetic operation: $0 + 0 - 4 = -4$. This calculation reveals that $y = -4$. This is the y-coordinate of our y-intercept. It's the precise value that tells us where the parabola intersects the y-axis. The solution, $y = -4$, is the key to locating the y-intercept on the coordinate plane.

Step 5: Express the $y$-intercept as a Coordinate Point

To fully express the y-intercept, we write it as a coordinate point. We know that the $x$-coordinate is zero, and we have calculated the $y$-coordinate to be -4. Therefore, the y-intercept is the point $(0, -4)$. This coordinate point provides a complete and unambiguous representation of the y-intercept on the graph. It's the precise location where the parabola crosses the y-axis, a crucial piece of information for understanding the function's behavior.

By meticulously following these steps, we have successfully determined the y-intercept of the quadratic function $y = -x^2 + 8x - 4$. This step-by-step approach not only provides the solution but also reinforces the underlying concepts and techniques involved in finding y-intercepts.

While mathematical calculations provide a precise understanding of the y-intercept, visualizing the parabola and its intersection with the y-axis can further enhance our comprehension. Graphing the quadratic function allows us to see the y-intercept in context, as a specific point on the curve. This visual representation can be invaluable for developing intuition about the relationship between the function's equation and its graphical form.

To visualize the parabola, we can plot a few key points. We already know the y-intercept is $(0, -4)$. We can also find the vertex of the parabola, which is the point where the parabola changes direction. The $x$-coordinate of the vertex is given by the formula $x = -b / 2a$, where $a$ and $b$ are the coefficients of the quadratic equation. In our case, $a = -1$ and $b = 8$, so the $x$-coordinate of the vertex is $x = -8 / (2 * -1) = 4$. Substituting $x = 4$ into the equation, we find the $y$-coordinate of the vertex: $y = -(4)^2 + 8(4) - 4 = -16 + 32 - 4 = 12$. Therefore, the vertex is located at the point $(4, 12)$.

Plotting the y-intercept $(0, -4)$ and the vertex $(4, 12)$, along with a few other points, allows us to sketch the parabola. The parabola opens downwards because the coefficient of the $x^2$ term is negative. We can clearly see the y-intercept as the point where the parabola crosses the y-axis. This visual confirmation reinforces our understanding of the y-intercept as a key feature of the parabola.

The y-intercept is not merely a point on a graph; it carries significant information about the function it represents. In various real-world applications, the y-intercept provides a crucial initial value or starting point. For instance, in a business scenario, if the quadratic function represents the profit of a company, the y-intercept might indicate the initial loss or investment before any sales are made. Similarly, in a physics context, if the function describes the height of a projectile, the y-intercept could represent the initial height from which the projectile was launched.

Understanding the y-intercept can also aid in sketching the graph of the quadratic function. Knowing the y-intercept, along with the vertex and the direction of the parabola's opening, provides a solid framework for visualizing the curve. This visual representation can be immensely helpful in solving problems, making predictions, and gaining a deeper understanding of the function's behavior.

In conclusion, finding the y-intercept of a quadratic function is a fundamental skill in mathematics. By substituting $x = 0$ into the equation and solving for $y$, we can determine the point where the parabola intersects the y-axis. This y-intercept provides valuable information about the function's behavior and can be visualized on the graph as a key anchor point. The y-intercept is not just a mathematical concept; it is a powerful tool for understanding and applying quadratic functions in various real-world scenarios.

The correct answer is A. (0, -4).