Finding Zeros And Y-intercept Of F(x) = -(x+1)(x-3)(x+2)

In the realm of mathematics, understanding the behavior of functions is paramount. Functions, the workhorses of mathematical modeling, describe relationships between variables, and their characteristics reveal valuable insights into the systems they represent. Among these characteristics, the zeros and y-intercept of a function hold particular significance. Zeros, also known as roots or x-intercepts, are the points where the function's graph intersects the x-axis, indicating the values of the independent variable (x) that make the function equal to zero. The y-intercept, on the other hand, is the point where the graph intersects the y-axis, representing the function's value when the independent variable is zero. These points provide crucial anchors for sketching the graph of a function and understanding its overall behavior.

In this article, we delve into the fascinating world of polynomial functions, specifically focusing on the function f(x) = -(x+1)(x-3)(x+2). Our primary objective is to determine the zeros and y-intercept of this function. By identifying these key points, we can gain a deeper understanding of the function's behavior and its graphical representation. The zeros will reveal where the function crosses the x-axis, providing insights into the function's roots, while the y-intercept will tell us the function's value when x is zero, giving us a starting point for sketching the graph. So, let's embark on this mathematical journey and unravel the mysteries of this polynomial function. Understanding zeros and y-intercepts is not just an academic exercise; it's a fundamental skill that empowers us to analyze and interpret real-world phenomena modeled by functions.

The zeros of a function are the values of x for which the function f(x) equals zero. In other words, they are the solutions to the equation f(x) = 0. For the function f(x) = -(x+1)(x-3)(x+2), we can find the zeros by setting the function equal to zero and solving for x. The beauty of this particular function lies in its factored form. The function is expressed as a product of factors, which simplifies the process of finding the zeros. According to the zero-product property, if the product of several factors is zero, then at least one of the factors must be zero. Applying this property to our function, we can set each factor equal to zero and solve for x.

Let's begin by setting the first factor, (x+1), equal to zero: x + 1 = 0. Solving for x, we subtract 1 from both sides of the equation, resulting in x = -1. This tells us that -1 is one of the zeros of the function. Next, we consider the second factor, (x-3), and set it equal to zero: x - 3 = 0. Adding 3 to both sides of the equation, we find x = 3. Therefore, 3 is another zero of the function. Finally, we examine the third factor, (x+2), and set it equal to zero: x + 2 = 0. Subtracting 2 from both sides, we obtain x = -2. This reveals that -2 is also a zero of the function. Thus, we have identified three zeros for the function f(x) = -(x+1)(x-3)(x+2): -1, 3, and -2. These zeros are the points where the graph of the function intersects the x-axis, providing us with crucial information about the function's behavior. Knowing the zeros allows us to sketch the graph more accurately and understand the intervals where the function is positive or negative. In essence, the zeros are the key to unlocking the function's secrets.

The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when the independent variable, x, is equal to zero. To find the y-intercept of the function f(x) = -(x+1)(x-3)(x+2), we simply substitute x = 0 into the function and evaluate the expression. This will give us the value of the function when x is zero, which is the y-coordinate of the y-intercept. Let's perform this calculation step by step. We begin by replacing every instance of x in the function with 0: f(0) = -(0+1)(0-3)(0+2).

Now, we simplify the expression inside the parentheses: f(0) = -(1)(-3)(2). Next, we multiply the numbers together, paying careful attention to the signs. The product of -1 and -3 is 3, so we have: f(0) = -(3)(2). Finally, we multiply 3 by 2, which gives us 6, and then apply the negative sign: f(0) = -6. Therefore, the y-intercept of the function f(x) = -(x+1)(x-3)(x+2) is -6. This means that the graph of the function intersects the y-axis at the point (0, -6). The y-intercept is a crucial point on the graph of a function because it tells us the function's value when x is zero. It serves as an anchor point for sketching the graph and helps us understand the function's behavior near the y-axis. In conjunction with the zeros, the y-intercept provides a comprehensive picture of the function's overall behavior and shape.

In this exploration of the function f(x) = -(x+1)(x-3)(x+2), we have successfully identified its zeros and y-intercept. By setting the function equal to zero and applying the zero-product property, we determined that the zeros of the function are -1, 3, and -2. These zeros represent the points where the graph of the function intersects the x-axis, providing us with crucial information about the function's roots and its behavior around these points. Furthermore, by substituting x = 0 into the function, we found that the y-intercept is -6. This y-intercept represents the point where the graph of the function intersects the y-axis, giving us the function's value when x is zero.

The zeros and y-intercept are fundamental characteristics of a function that provide valuable insights into its behavior and graphical representation. The zeros tell us where the function crosses the x-axis, while the y-intercept tells us where it crosses the y-axis. These points serve as anchors for sketching the graph of the function and understanding its overall shape. In the case of the function f(x) = -(x+1)(x-3)(x+2), knowing the zeros (-1, 3, and -2) and the y-intercept (-6) allows us to create a more accurate and informative sketch of the function's graph. This understanding is not only crucial for mathematical analysis but also for applying functions to model real-world phenomena. By mastering the techniques for finding zeros and y-intercepts, we equip ourselves with powerful tools for understanding and interpreting the mathematical world around us. The combination of zeros and y-intercept paints a comprehensive picture of the function's behavior, allowing us to analyze its trends, predict its values, and make informed decisions based on the mathematical model it represents.

The zeros of the function f(x) = -(x+1)(x-3)(x+2) are -1, 3, and -2, and the y-intercept of the function is located at (0, -6).