Finding The Y-intercept Of The Function F(x)=(x-2)^2(x+3)(x+1)

In the realm of mathematics, polynomial functions play a pivotal role, serving as fundamental building blocks for modeling various phenomena. Understanding the characteristics of these functions, such as their intercepts, is crucial for analyzing their behavior and applications. The $y$-intercept, in particular, holds significant importance as it represents the point where the function's graph intersects the $y$-axis, providing valuable insights into the function's initial value or starting point.

Decoding the Significance of the $y$-intercept

The $y$-intercept, also known as the vertical intercept, is the point where the graph of a function crosses the $y$-axis. At this point, the $x$-coordinate is always zero. The $y$-intercept provides valuable information about the function's behavior, including:

• Initial Value: In many real-world applications, the $y$-intercept represents the initial value of the quantity being modeled by the function. For instance, if the function represents the population of a city over time, the $y$-intercept would represent the initial population at time zero.
• Starting Point: The $y$-intercept can be considered the starting point of the function's graph. It helps visualize the function's behavior as it moves away from the $y$-axis.
• Constant Term: The $y$-intercept is directly related to the constant term of the polynomial function. In the standard form of a polynomial, the constant term is the value of the function when $x$ is zero, which corresponds to the $y$-intercept.

Unmasking the $y$-intercept of $f(x)=(x-2)^2(x+3)(x+1)$

To determine the $y$-intercept of the given function, $f(x)=(x-2)^2(x+3)(x+1)$, we need to find the value of the function when $x=0$. This is because the $y$-axis is defined by the equation $x=0$. Substituting $x=0$ into the function, we get:

$f(0) = (0-2)^2(0+3)(0+1) = (-2)^2(3)(1) = 4 imes 3 imes 1 = 12$

Therefore, the $y$-intercept of the function $f(x)=(x-2)^2(x+3)(x+1)$ is 12. This means that the graph of the function intersects the $y$-axis at the point $(0, 12)$.

Step-by-Step Calculation of the $y$-intercept

To further clarify the process of finding the $y$-intercept, let's break down the calculation into a series of steps:

1. Substitute $x=0$ into the function: This is the fundamental step in finding the $y$-intercept. We replace every instance of $x$ in the function's expression with the value 0.
2. Simplify the expression: After substituting $x=0$, we simplify the resulting expression by performing the necessary arithmetic operations, such as addition, subtraction, multiplication, and exponentiation.
3. Evaluate the result: The final result of the simplified expression is the $y$-intercept of the function. It represents the $y$-coordinate of the point where the graph intersects the $y$-axis.

In the case of $f(x)=(x-2)^2(x+3)(x+1)$:

1. Substitute $x=0$: $f(0) = (0-2)^2(0+3)(0+1)$
2. Simplify the expression: $f(0) = (-2)^2(3)(1) = 4 imes 3 imes 1$
3. Evaluate the result: $f(0) = 12$

Thus, the $y$-intercept is 12.

Visualizing the $y$-intercept on the Graph

The $y$-intercept can be easily visualized on the graph of the function. It is the point where the graph crosses the vertical $y$-axis. In the case of $f(x)=(x-2)^2(x+3)(x+1)$, the graph would intersect the $y$-axis at the point $(0, 12)$. This point provides a visual reference for the function's behavior near the $y$-axis.

The graph of the function can be plotted using various graphing tools or software. By observing the graph, we can confirm that it indeed crosses the $y$-axis at the point $(0, 12)$, reinforcing our calculated result.

Connecting the $y$-intercept to the Constant Term

There is a direct connection between the $y$-intercept and the constant term of a polynomial function. To understand this connection, let's first expand the given function:

$f(x) = (x-2)^2(x+3)(x+1) = (x^2 - 4x + 4)(x^2 + 4x + 3)$

Expanding further, we get:

$f(x) = x^4 + 4x^3 + 3x^2 - 4x^3 - 16x^2 - 12x + 4x^2 + 16x + 12$

Simplifying the expression, we obtain the standard form of the polynomial:

$f(x) = x^4 - 9x^2 + 4x + 12$

Notice that the constant term in the standard form is 12, which is the same as the $y$-intercept we calculated earlier. This observation highlights the direct relationship between the $y$-intercept and the constant term of a polynomial function.

In general, for any polynomial function in the standard form:

$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

the constant term $a_0$ represents the $y$-intercept of the function. This is because when $x=0$, all the terms with $x$ become zero, leaving only the constant term $a_0$.

Real-World Applications of the $y$-intercept

The $y$-intercept finds applications in various real-world scenarios. Here are a couple of examples:

• Population Growth: If a function models the population of a city over time, the $y$-intercept represents the initial population of the city at the starting time.
• Financial Investments: If a function models the growth of an investment over time, the $y$-intercept represents the initial investment amount.
• Physics: In physics, if a function represents the position of an object over time, the $y$-intercept represents the initial position of the object.

These examples illustrate how the $y$-intercept provides valuable information about the initial state or starting point of a quantity being modeled by a function.

Conclusion: The Significance of the $y$-intercept

In conclusion, the $y$-intercept is a crucial characteristic of a function, providing valuable insights into its behavior and applications. For the polynomial function $f(x)=(x-2)^2(x+3)(x+1)$, the $y$-intercept is 12, indicating that the graph of the function intersects the $y$-axis at the point $(0, 12)$. This understanding of the $y$-intercept enhances our ability to analyze and interpret polynomial functions in various mathematical and real-world contexts.

By mastering the concept of the $y$-intercept and its calculation, you gain a powerful tool for understanding and working with polynomial functions. Remember, the $y$-intercept is not just a number; it's a key piece of information that unlocks the secrets of a function's behavior and its connection to the real world.