Finding The Y-intercept Of F(x)=(x-6)(x-2) A Step-by-Step Guide

In the realm of quadratic functions, understanding key features like the $y$-intercept is crucial for grasping the behavior and graph of the function. This article will delve into the process of finding the $y$-intercept of the quadratic function $f(x) = (x-6)(x-2)$. We'll explore the underlying concepts, walk through the calculations, and discuss why the $y$-intercept is a significant characteristic of quadratic functions. The question at hand is: What is the $y$-intercept of the quadratic function $f(x) = (x-6)(x-2)$?

Understanding Quadratic Functions and the $y$-intercept

To begin, let's establish a firm understanding of quadratic functions. Quadratic functions are polynomial functions of the second degree, generally expressed in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a e 0$. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if $a > 0$ and downwards if $a < 0$. The $y$-intercept, in particular, is the point where the parabola intersects the $y$-axis. This intersection occurs when the $x$-coordinate is zero. Therefore, to find the $y$-intercept, we simply need to evaluate the function at $x = 0$.

The significance of the $y$-intercept lies in its ability to provide a starting point for understanding the function's behavior. It tells us the value of the function when the input is zero, which can be interpreted as the initial value in many real-world applications. For instance, if the quadratic function models the trajectory of a projectile, the $y$-intercept might represent the initial height of the projectile. In the context of business and economics, it could represent the fixed costs before any units are produced or sold. Thus, identifying the $y$-intercept is not just a mathematical exercise but also a practical skill with far-reaching implications.

The given function, $f(x) = (x-6)(x-2)$, is presented in factored form. While this form directly reveals the $x$-intercepts (also known as roots or zeros) of the function, it doesn't immediately show the $y$-intercept. To find the $y$-intercept, we need to substitute $x = 0$ into the function. The factored form, however, provides an advantage in calculation simplicity, especially when compared to the standard form $ax^2 + bx + c$. By substituting $x = 0$ into the factored form, we can efficiently determine the value of $f(0)$, which corresponds to the $y$-coordinate of the $y$-intercept. This process underscores the importance of understanding different forms of quadratic functions and how each form offers unique insights into the function's characteristics.

Finding the $y$-intercept of $f(x)=(x-6)(x-2)$

Now, let's apply the concept to the specific quadratic function $f(x) = (x-6)(x-2)$. To find the $y$-intercept, we need to determine the value of $f(x)$ when $x = 0$. This means substituting $0$ for $x$ in the function's equation:

$f(0) = (0 - 6)(0 - 2)$

This substitution simplifies the expression, allowing us to perform the arithmetic operations directly. We first evaluate the expressions within the parentheses:

$f(0) = (-6)(-2)$

Next, we multiply the two negative numbers. Recall that the product of two negative numbers is a positive number. Thus,

$f(0) = 12$

This result tells us that when $x = 0$, the value of the function $f(x)$ is $12$. Therefore, the $y$-intercept is the point where the graph of the function intersects the $y$-axis, which occurs at the coordinate $(0, 12)$. The $y$-intercept, $(0, 12)$, signifies that the parabola crosses the $y$-axis at the point where $y$ is equal to $12$. This point is a crucial reference when sketching the graph of the parabola or analyzing its behavior.

In the context of real-world applications, this $y$-intercept could represent an initial condition or a starting value. For instance, if $f(x)$ models the height of a ball thrown in the air, the $y$-intercept $(0, 12)$ might indicate that the ball was initially thrown from a height of $12$ units. Understanding the $y$-intercept allows us to anchor the function in a meaningful way within the given scenario. Furthermore, the process of finding the $y$-intercept reinforces the fundamental concept of function evaluation, which is a cornerstone of mathematical analysis.

Analyzing the Given Options

Having calculated the $y$-intercept to be $(0, 12)$, let's examine the provided options to identify the correct answer. The options are:

A. $(2, 0)$ B. $(0, 12)$ C. $(-8, 0)$ D. $(0, -6)$

Comparing our calculated $y$-intercept, $(0, 12)$, with the given options, we can clearly see that option B, $(0, 12)$, matches our result. Options A and C, $(2, 0)$ and $(-8, 0)$, represent the $x$-intercepts of the function, also known as the roots or zeros. These are the points where the graph intersects the $x$-axis, which are found by setting $f(x) = 0$ and solving for $x$. In this case, the $x$-intercepts are indeed $x = 6$ and $x = 2$, corresponding to the factors $(x - 6)$ and $(x - 2)$ in the function's equation.

Option D, $(0, -6)$, is incorrect because it does not satisfy the condition $f(0) = 12$. This option represents a point on the $y$-axis, but not the specific point where the given quadratic function intersects it. The distinction between the $x$-intercepts and the $y$-intercept is fundamental. The $x$-intercepts are the solutions to the equation $f(x) = 0$, while the $y$-intercept is the value of $f(x)$ when $x = 0$. Confusing these concepts can lead to errors in identifying key features of the graph and in applying the function to real-world problems.

The process of analyzing these options highlights the importance of careful calculation and conceptual clarity. It's not enough to simply perform the calculation; one must also understand what the result represents in the context of the function and its graph. Recognizing the significance of the $y$-intercept as the point of intersection with the $y$-axis, and distinguishing it from the $x$-intercepts, is crucial for accurate interpretation and problem-solving.

Conclusion

In conclusion, the $y$-intercept of the quadratic function $f(x) = (x-6)(x-2)$ is $(0, 12)$. We arrived at this answer by substituting $x = 0$ into the function and evaluating the expression. The $y$-intercept is a key feature of a quadratic function, representing the point where the parabola intersects the $y$-axis. It provides valuable information about the function's behavior and can be interpreted as an initial value in various applications.

Understanding how to find the $y$-intercept is essential for analyzing quadratic functions and their graphs. This process involves recognizing the significance of the $y$-intercept, applying the correct substitution, and interpreting the result in the context of the function. By mastering these steps, one can confidently identify and utilize the $y$-intercept in a wide range of mathematical and real-world scenarios. Moreover, this exercise underscores the importance of distinguishing the $y$-intercept from other key features, such as the $x$-intercepts, and recognizing the unique information each provides about the function.

The ability to accurately determine the $y$-intercept is a fundamental skill in algebra and calculus, serving as a building block for more advanced concepts. It exemplifies the interplay between algebraic manipulation and graphical interpretation, which is a cornerstone of mathematical understanding. As we've seen, the $y$-intercept is more than just a point on a graph; it's a meaningful value that can provide insights into the function's behavior and its applications. Thus, a solid grasp of this concept is crucial for success in mathematics and related fields.