Find The $y$-coordinate Of The $y$-intercept Of The Polynomial Function Defined Below.$f(x) = 2(x - 6)$

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Introduction

In mathematics, the $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is a crucial concept in understanding the behavior of functions, particularly polynomial functions. In this article, we will focus on finding the $y$-intercept of a polynomial function defined by $f(x) = 2(x - 6)$.

Understanding the $y$-Intercept

The $y$-intercept of a function is the value of $y$ when $x = 0$. In other words, it is the point on the graph where the $x$-coordinate is zero. To find the $y$-intercept of a function, we need to substitute $x = 0$ into the function and solve for $y$.

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

To find the $y$-intercept of the function $f(x) = 2(x - 6)$, we need to substitute $x = 0$ into the function and solve for $y$. We can do this by following these steps:

  1. Substitute $x = 0$ into the function: $f(0) = 2(0 - 6)$
  2. Simplify the expression: $f(0) = 2(-6)$
  3. Evaluate the expression: $f(0) = -12$

Therefore, the $y$-intercept of the function $f(x) = 2(x - 6)$ is $-12$.

Interpretation of the $y$-Intercept

The $y$-intercept of a function represents the starting point of the graph. It is the point where the graph begins to rise or fall. In the case of the function $f(x) = 2(x - 6)$, the $y$-intercept is $-12$. This means that the graph of the function starts at the point $(0, -12)$ and rises or falls from there.

Real-World Applications of the $y$-Intercept

The $y$-intercept of a function has many real-world applications. For example, in economics, the $y$-intercept of a demand curve represents the starting point of the demand for a product. In physics, the $y$-intercept of a velocity-time graph represents the initial velocity of an object.

Conclusion

In conclusion, finding the $y$-intercept of a polynomial function is a crucial concept in understanding the behavior of functions. By substituting $x = 0$ into the function and solving for $y$, we can find the $y$-intercept of a function. In this article, we found the $y$-intercept of the function $f(x) = 2(x - 6)$ to be $-12$. We also discussed the interpretation and real-world applications of the $y$-intercept.

Additional Examples

Here are a few additional examples of finding the $y$-intercept of a polynomial function:

  • Example 1: Find the $y$-intercept of the function $f(x) = 3(x + 2)$.
    • Substitute $x = 0$ into the function: $f(0) = 3(0 + 2)$
    • Simplify the expression: $f(0) = 3(2)$
    • Evaluate the expression: $f(0) = 6$
  • Example 2: Find the $y$-intercept of the function $f(x) = 4(x - 3)$.
    • Substitute $x = 0$ into the function: $f(0) = 4(0 - 3)$
    • Simplify the expression: $f(0) = 4(-3)$
    • Evaluate the expression: $f(0) = -12$

Practice Problems

Here are a few practice problems to help you find the $y$-intercept of a polynomial function:

  • Find the $y$-intercept of the function $f(x) = 2(x + 4)$.
  • Find the $y$-intercept of the function $f(x) = 3(x - 2)$.
  • Find the $y$-intercept of the function $f(x) = 4(x + 1)$.

Solutions

Here are the solutions to the practice problems:

  • Problem 1: Find the $y$-intercept of the function $f(x) = 2(x + 4)$.
    • Substitute $x = 0$ into the function: $f(0) = 2(0 + 4)$
    • Simplify the expression: $f(0) = 2(4)$
    • Evaluate the expression: $f(0) = 8$
  • Problem 2: Find the $y$-intercept of the function $f(x) = 3(x - 2)$.
    • Substitute $x = 0$ into the function: $f(0) = 3(0 - 2)$
    • Simplify the expression: $f(0) = 3(-2)$
    • Evaluate the expression: $f(0) = -6$
  • Problem 3: Find the $y$-intercept of the function $f(x) = 4(x + 1)$.
    • Substitute $x = 0$ into the function: $f(0) = 4(0 + 1)$
    • Simplify the expression: $f(0) = 4(1)$
    • Evaluate the expression: $f(0) = 4$
      Q&A: Finding the $y$-Intercept of a Polynomial Function ===========================================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about finding the $y$-intercept of a polynomial function.

Q: What is the $y$-intercept of a function?

A: The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. It is the value of $y$ when $x = 0$.

Q: How do I find the $y$-intercept of a polynomial function?

A: To find the $y$-intercept of a polynomial function, you need to substitute $x = 0$ into the function and solve for $y$. This can be done by following these steps:

  1. Substitute $x = 0$ into the function: Replace $x$ with $0$ in the function.
  2. Simplify the expression: Simplify the expression by combining like terms.
  3. Evaluate the expression: Evaluate the expression to find the value of $y$.

Q: What is the $y$-intercept of the function $f(x) = 2(x - 6)$?

A: To find the $y$-intercept of the function $f(x) = 2(x - 6)$, we need to substitute $x = 0$ into the function and solve for $y$. We can do this by following these steps:

  1. Substitute $x = 0$ into the function: $f(0) = 2(0 - 6)$
  2. Simplify the expression: $f(0) = 2(-6)$
  3. Evaluate the expression: $f(0) = -12$

Therefore, the $y$-intercept of the function $f(x) = 2(x - 6)$ is $-12$.

Q: How do I interpret the $y$-intercept of a function?

A: The $y$-intercept of a function represents the starting point of the graph. It is the point where the graph begins to rise or fall. In the case of the function $f(x) = 2(x - 6)$, the $y$-intercept is $-12$. This means that the graph of the function starts at the point $(0, -12)$ and rises or falls from there.

Q: What are some real-world applications of the $y$-intercept of a function?

A: The $y$-intercept of a function has many real-world applications. For example, in economics, the $y$-intercept of a demand curve represents the starting point of the demand for a product. In physics, the $y$-intercept of a velocity-time graph represents the initial velocity of an object.

Q: How do I find the $y$-intercept of a function with a negative exponent?

A: To find the $y$-intercept of a function with a negative exponent, you need to follow the same steps as before. However, you need to be careful when simplifying the expression. For example, if the function is $f(x) = 2(x^{-1})$, you need to substitute $x = 0$ into the function and solve for $y$. We can do this by following these steps:

  1. Substitute $x = 0$ into the function: $f(0) = 2(0^{-1})$
  2. Simplify the expression: $f(0) = 2(\infty)$
  3. Evaluate the expression: $f(0) = \infty$

Therefore, the $y$-intercept of the function $f(x) = 2(x^{-1})$ is $\infty$.

Q: What is the difference between the $y$-intercept and the $x$-intercept of a function?

A: The $y$-intercept of a function is the point at which the graph of the function intersects the $y$-axis. The $x$-intercept of a function is the point at which the graph of the function intersects the $x$-axis. In other words, the $y$-intercept is the value of $y$ when $x = 0$, while the $x$-intercept is the value of $x$ when $y = 0$.

Q: How do I find the $x$-intercept of a function?

A: To find the $x$-intercept of a function, you need to substitute $y = 0$ into the function and solve for $x$. This can be done by following these steps:

  1. Substitute $y = 0$ into the function: Replace $y$ with $0$ in the function.
  2. Simplify the expression: Simplify the expression by combining like terms.
  3. Evaluate the expression: Evaluate the expression to find the value of $x$.

Conclusion

In conclusion, finding the $y$-intercept of a polynomial function is a crucial concept in understanding the behavior of functions. By substituting $x = 0$ into the function and solving for $y$, we can find the $y$-intercept of a function. We also discussed the interpretation and real-world applications of the $y$-intercept.