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

In the realm of quadratic equations, the $y$-intercept holds a crucial position, marking the point where the parabola gracefully intersects the vertical $y$-axis. For the given quadratic equation, $y = -x^2 + 8x - 4$, our mission is to pinpoint this elusive $y$-intercept. To embark on this mathematical journey, we'll delve into the fundamental concept of $y$-intercepts and employ a systematic approach to unveil its coordinates.

Understanding the Essence of $y$-intercepts

The $y$-intercept, in its simplest form, represents the point at which a graph crosses the $y$-axis. This intersection occurs when the $x$-coordinate is precisely zero. Geometrically, it's the point where the curve pierces the vertical axis, providing a vital anchor point for sketching or analyzing the graph. In the context of quadratic equations, the $y$-intercept reveals the value of the quadratic function when $x$ is set to zero. This value holds significant meaning, often representing the initial state or starting point of the phenomenon modeled by the equation. For instance, in projectile motion, the $y$-intercept could signify the initial height of the projectile when launched. Similarly, in financial models, it might represent the initial investment or starting capital.

Keywords: y-intercept, graph, quadratic equation, y-axis, x-coordinate, function, initial value, parabola, intersection, vertical axis.

The Quest for the $y$-intercept: A Step-by-Step Approach

To determine the $y$-intercept of the quadratic equation $y = -x^2 + 8x - 4$, we'll employ a straightforward yet elegant method. We'll set the $x$-coordinate to zero and then evaluate the equation to find the corresponding $y$-coordinate. This process stems directly from the definition of the $y$-intercept as the point where the graph intersects the $y$-axis, which inherently occurs when $x = 0$. By substituting $x = 0$ into the equation, we effectively isolate the $y$-value at this specific point, revealing the $y$-coordinate of the intercept. This approach is not only mathematically sound but also intuitively clear, providing a direct path to uncovering the $y$-intercept.

1. Setting $x$ to Zero: The cornerstone of our approach lies in substituting $x = 0$ into the given equation. This substitution allows us to focus solely on the $y$-value when the graph intersects the $y$-axis. Mathematically, this translates to replacing every instance of $x$ in the equation with the value 0.
2. Evaluating the Equation: Once we've made the substitution, we simplify the equation to solve for $y$. This involves performing the arithmetic operations, such as squaring, multiplication, and addition/subtraction, in the correct order. The result will be a numerical value representing the $y$-coordinate of the $y$-intercept.
3. Expressing the $y$-intercept: Finally, we express the $y$-intercept as an ordered pair $(0, y)$, where the $x$-coordinate is 0 (by definition) and the $y$-coordinate is the value we calculated in the previous step. This ordered pair provides a complete representation of the $y$-intercept, clearly indicating its position on the coordinate plane.

Keywords: substituting, equation, solving, y-coordinate, ordered pair, coordinate plane, step-by-step, mathematical method, simplification, arithmetic operations.

Unveiling the Solution: A Detailed Calculation

Now, let's put our method into action and calculate the $y$-intercept of the equation $y = -x^2 + 8x - 4$. We'll meticulously follow the steps outlined above to arrive at the solution.

1. Setting $x$ to Zero: We begin by substituting $x = 0$ into the equation:

   $y = -(0)^2 + 8(0) - 4$

   This substitution replaces every $x$ with 0, setting the stage for isolating the $y$-value.

2. Evaluating the Equation: Next, we simplify the equation by performing the arithmetic operations:

   $y = -0 + 0 - 4$ $y = -4$

   This simplification reveals that when $x = 0$, the value of $y$ is $-4$.

3. Expressing the $y$-intercept: Finally, we express the $y$-intercept as an ordered pair:

   $(0, -4)$

   This ordered pair represents the point where the parabola intersects the $y$-axis. The $y$-intercept is located at the point $(0, -4)$ on the coordinate plane.

Therefore, the $y$-intercept of the graph of $y = -x^2 + 8x - 4$ is (0, -4). This point marks the intersection of the parabola with the vertical axis, providing a crucial reference point for understanding the behavior of the quadratic function.

Keywords: calculation, substitution, simplification, arithmetic operations, ordered pair, coordinate plane, intersection, vertical axis, reference point, quadratic function.

Why the Other Options Are Incorrect: A Process of Elimination

To solidify our understanding, let's examine why the other options provided are incorrect. This process of elimination will not only reinforce the correct solution but also deepen our grasp of the underlying concepts.

A. (0, 4): This option suggests that the $y$-intercept is at $y = 4$. However, our calculations clearly showed that when $x = 0$, $y = -4$, not 4. Therefore, this option is incorrect.

B. (0, 0): This option implies that the parabola passes through the origin (0, 0). However, our calculations demonstrated that the $y$-intercept is at $(0, -4)$, not the origin. Thus, this option is also incorrect.

C. (0, -∞) and D. (0, -∞): These options are nonsensical in the context of $y$-intercepts. The $y$-intercept is a specific point on the $y$-axis, represented by a finite numerical value. Negative infinity ($-\infty$) is not a specific value but rather a concept representing unbounded decrease. Therefore, these options are invalid.

By systematically eliminating these incorrect options, we further strengthen our confidence in the correct answer, which is (0, -4). This process highlights the importance of careful calculation and conceptual understanding in solving mathematical problems.

Keywords: process of elimination, incorrect options, calculations, conceptual understanding, finite numerical value, negative infinity, mathematical problems, origin, y-axis, specific point.

The Significance of the $y$-intercept in Quadratic Functions

The $y$-intercept holds a special significance in the analysis of quadratic functions. It represents the value of the function when $x = 0$, often providing a crucial starting point or initial condition for the scenario being modeled. In various real-world applications, the $y$-intercept can offer valuable insights.

For instance, in the context of projectile motion, the $y$-intercept might represent the initial height of an object launched into the air. In financial modeling, it could signify the initial investment or starting capital. In population growth models, the $y$-intercept could indicate the initial population size. These examples illustrate the practical relevance of the $y$-intercept in interpreting quadratic functions within real-world contexts.

Furthermore, the $y$-intercept, along with the vertex and the $x$-intercepts, helps to define the shape and position of the parabola. These key points provide a framework for sketching the graph of the quadratic function and understanding its overall behavior. The $y$-intercept, in particular, anchors the parabola to the $y$-axis, serving as a crucial reference point.

In conclusion, the $y$-intercept is not merely a point on a graph; it's a vital piece of information that can unlock deeper insights into the meaning and behavior of quadratic functions. Its significance extends beyond the mathematical realm, finding applications in diverse fields such as physics, finance, and biology.

Keywords: significance, quadratic functions, real-world applications, projectile motion, financial modeling, population growth, parabola shape, graph sketching, overall behavior, crucial reference point.

In summary, we embarked on a mathematical journey to unveil the $y$-intercept of the quadratic equation $y = -x^2 + 8x - 4$. Through a systematic approach, we substituted $x = 0$ into the equation, meticulously evaluated the expression, and arrived at the solution: (0, -4). This point represents the graceful intersection of the parabola with the $y$-axis. We also explored why the other options were incorrect, reinforcing our understanding of the concept.

Furthermore, we delved into the significance of the $y$-intercept in quadratic functions, highlighting its role as a starting point, initial condition, and a key reference point for sketching graphs and interpreting real-world phenomena. The $y$-intercept is not merely a mathematical artifact; it's a gateway to understanding the behavior and implications of quadratic relationships.

This exploration underscores the importance of a solid grasp of fundamental concepts in mathematics. By mastering techniques like finding the $y$-intercept, we empower ourselves to analyze and interpret the world around us through the lens of mathematical modeling.

Keywords: mathematical journey, systematic approach, equation evaluation, parabola intersection, graph sketching, real-world phenomena, fundamental concepts, mathematical modeling, analytical skills, quadratic relationships.