Which Point Represents The X-intercept Of The Quadratic Function F(x) = (x + 6)(x - 3)? Options: A. (0, 6) B. (0, -6) C. (6, 0) D. (-6, 0)

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In mathematics, particularly in algebra, understanding the characteristics of functions is crucial for problem-solving and real-world applications. Among these characteristics, intercepts hold significant importance. The x-intercepts of a function are the points where the graph of the function intersects the x-axis. These points are particularly valuable because they represent the solutions or roots of the function. In the context of quadratic functions, finding the x-intercepts is a common task that provides insights into the behavior and properties of the parabola represented by the function.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 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 x-intercepts of a quadratic function are the points where the parabola intersects the x-axis. At these points, the value of the function, f(x), is zero. Therefore, to find the x-intercepts, we need to solve the quadratic equation ax² + bx + c = 0.

Methods for Finding X-Intercepts

There are several methods to find the x-intercepts of a quadratic function, including:

  1. Factoring: If the quadratic expression can be factored, setting each factor equal to zero will yield the x-intercepts. This method is straightforward when the quadratic expression has integer roots.

  2. Quadratic Formula: The quadratic formula is a universal method for finding the roots of any quadratic equation. The formula is given by:

    x = (-b ± √(b² - 4ac)) / (2a)

    This formula provides the x-intercepts regardless of whether the roots are real or complex.

  3. Completing the Square: This method involves rewriting the quadratic equation in a form that allows for easy extraction of the roots. It is particularly useful for understanding the vertex form of the quadratic function.

Given Function: f(x) = (x + 6)(x - 3)

Let's consider the given quadratic function: f(x) = (x + 6)(x - 3). This function is already in factored form, which makes finding the x-intercepts relatively simple. The x-intercepts occur where f(x) = 0. Therefore, we need to solve the equation:

(x + 6)(x - 3) = 0

To solve this equation, we set each factor equal to zero:

  • x + 6 = 0
  • x - 3 = 0

Solving these equations gives us:

  • x = -6
  • x = 3

Therefore, the x-intercepts are the points where the graph of the function intersects the x-axis, which occur at x = -6 and x = 3. These points are represented as coordinates (-6, 0) and (3, 0).

Analyzing the Options

Now, let's analyze the given options to determine which point is an x-intercept of the function f(x) = (x + 6)(x - 3).

  • A. (0, 6): This point has an x-coordinate of 0 and a y-coordinate of 6. This is a y-intercept, not an x-intercept.
  • B. (0, -6): This point also has an x-coordinate of 0 and a y-coordinate of -6. This is another y-intercept.
  • C. (6, 0): This point has an x-coordinate of 6 and a y-coordinate of 0. While the y-coordinate is 0, the x-coordinate does not match the x-intercepts we found (-6 and 3).
  • D. (-6, 0): This point has an x-coordinate of -6 and a y-coordinate of 0. This matches one of the x-intercepts we calculated.

Therefore, the correct answer is D. (-6, 0).

The Significance of X-Intercepts

X-intercepts are crucial in various mathematical and real-world contexts. Here are some reasons why they are significant:

  1. Roots of the Equation: The x-intercepts of a function are the roots or solutions of the equation f(x) = 0. These roots are the values of x that make the function equal to zero.
  2. Graphing Functions: X-intercepts are essential points for graphing functions. They indicate where the graph crosses the x-axis, providing a visual representation of the function's behavior.
  3. Real-World Applications: In real-world applications, x-intercepts can represent significant values. For example, in a projectile motion problem, the x-intercepts can represent the points where the projectile lands on the ground.
  4. Optimization Problems: X-intercepts can help in solving optimization problems. They can represent the points where a function reaches its maximum or minimum value.

Practical Examples

To further illustrate the concept of x-intercepts, let's consider some practical examples:

  1. Projectile Motion: Suppose a ball is thrown into the air, and its height h(t) at time t is given by the quadratic function h(t) = -16t² + 80t. The x-intercepts of this function represent the times when the ball hits the ground. To find these intercepts, we set h(t) = 0 and solve for t:

    -16t² + 80t = 0

    t(-16t + 80) = 0

    t = 0 or -16t + 80 = 0

    t = 0 or t = 5

    The x-intercepts are t = 0 and t = 5, which means the ball is initially on the ground at t = 0 and lands back on the ground at t = 5 seconds.

  2. Business Applications: In business, quadratic functions can be used to model profit or revenue. For example, a company's profit P(x) from selling x units of a product might be modeled by a quadratic function. The x-intercepts of this function represent the break-even points, where the profit is zero. Knowing the break-even points can help the company make informed decisions about pricing and production levels.

Common Mistakes to Avoid

When finding x-intercepts, it's essential to avoid common mistakes that can lead to incorrect answers. Here are some common pitfalls to watch out for:

  1. Confusing X-Intercepts with Y-Intercepts: X-intercepts occur where y = 0, while y-intercepts occur where x = 0. It's crucial to substitute the correct value to find the intercepts.
  2. Incorrect Factoring: Factoring quadratic expressions correctly is essential for finding x-intercepts. Double-check your factoring to ensure it is accurate.
  3. Misapplying the Quadratic Formula: The quadratic formula is a powerful tool, but it must be applied correctly. Ensure that you substitute the values of a, b, and c accurately and follow the order of operations.
  4. Forgetting the ± Sign: When using the quadratic formula, remember to include both the positive and negative square roots to find both x-intercepts.

Conclusion

Finding the x-intercepts of a quadratic function is a fundamental skill in algebra with various applications in mathematics and real-world scenarios. By understanding the different methods for finding x-intercepts and avoiding common mistakes, you can confidently solve related problems. In the case of the function f(x) = (x + 6)(x - 3), the x-intercept is indeed (-6, 0), as demonstrated by setting the function equal to zero and solving for x. Mastering the concept of x-intercepts will undoubtedly enhance your problem-solving abilities and deepen your understanding of quadratic functions.