Finding The X-Intercept Of F(x) = (x + 6)(x - 3) A Step-by-Step Guide
In the realm of quadratic functions, identifying the x-intercepts is a fundamental concept. These intercepts, also known as roots or zeros, are the points where the parabola intersects the x-axis. Understanding how to find these points is crucial for graphing quadratic functions and solving related problems. In this article, we will delve into the methods for determining the x-intercepts of a quadratic function, focusing on the factored form, and illustrate the process with the example function f(x) = (x + 6)(x - 3). We will also dissect the given options to pinpoint the correct x-intercept.
Understanding X-Intercepts
X-intercepts are the points where the graph of a function crosses the x-axis. At these points, the y-value (or the function value, f(x)) is zero. This is because any point on the x-axis has a y-coordinate of 0. Therefore, to find the x-intercepts of a function, we set f(x) equal to zero and solve for x. This process essentially identifies the x-values that make the function's output zero, which corresponds to the points where the graph touches or crosses the x-axis.
For a quadratic function in the standard form of f(x) = ax² + bx + c, finding the x-intercepts can involve factoring, using the quadratic formula, or completing the square. However, when the quadratic function is given in factored form, such as in our example, the process simplifies significantly. The factored form directly reveals the roots of the equation, making it easier to identify the x-intercepts.
Finding X-Intercepts from Factored Form
When a quadratic function is expressed in factored form, such as f(x) = (x - r₁) (x - r₂), the x-intercepts are readily apparent. The values r₁ and r₂ represent the roots of the equation, which are the x-values that make the function equal to zero. To find these roots, we set each factor equal to zero and solve for x.
This method leverages the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. In the context of quadratic functions, this means that if (x - r₁) (x - r₂) = 0, then either (x - r₁) = 0 or (x - r₂) = 0 (or both). Solving these equations gives us x = r₁ and x = r₂, which are the x-coordinates of the x-intercepts.
Each x-intercept corresponds to a point on the graph of the quadratic function where the parabola intersects the x-axis. These points are represented as (r₁, 0) and (r₂, 0), where the y-coordinate is zero, as explained earlier. Understanding this direct relationship between the factored form and the x-intercepts is crucial for quickly analyzing and graphing quadratic functions.
Applying the Concept to f(x) = (x + 6)(x - 3)
Now, let's apply this method to the given quadratic function, f(x) = (x + 6)(x - 3), to find its x-intercepts. Following the principle of setting each factor equal to zero, we have two equations to solve:
- x + 6 = 0
- x - 3 = 0
Solving the first equation, x + 6 = 0, we subtract 6 from both sides to isolate x: x = -6. This tells us that one of the x-intercepts occurs when x is -6.
Solving the second equation, x - 3 = 0, we add 3 to both sides to isolate x: x = 3. This indicates that another x-intercept occurs when x is 3.
Therefore, the x-intercepts of the function f(x) = (x + 6)(x - 3) are x = -6 and x = 3. These x-values correspond to the points (-6, 0) and (3, 0) on the graph of the function. These are the points where the parabola intersects the x-axis. By finding these intercepts, we gain valuable information about the behavior and graph of the quadratic function.
Analyzing the Options
Now that we have determined the x-intercepts of the function f(x) = (x + 6)(x - 3) to be (-6, 0) and (3, 0), we can analyze the given options to identify the correct answer. The options are:
A. (0, -6) B. (6, 0) C. (-6, 0) D. (0, 6)
Option A, (0, -6), represents a point where x = 0 and y = -6. This is a y-intercept, not an x-intercept, as the x-value is zero, not the y-value. Y-intercepts occur where the graph crosses the y-axis, which is a different concept from x-intercepts.
Option B, (6, 0), represents a point where x = 6 and y = 0. This point lies on the x-axis, but it is not one of the x-intercepts we calculated. Our calculations showed that the x-intercepts occur at x = -6 and x = 3, not at x = 6.
Option C, (-6, 0), represents a point where x = -6 and y = 0. This point aligns perfectly with one of the x-intercepts we calculated. When x is -6, the function f(x) equals zero, making this point an x-intercept.
Option D, (0, 6), represents a point where x = 0 and y = 6. Similar to option A, this is a y-intercept, not an x-intercept. The x-value is zero, indicating that the point lies on the y-axis.
Therefore, by comparing the calculated x-intercepts with the given options, we can confidently identify the correct answer.
Conclusion
In conclusion, the x-intercepts of the quadratic function f(x) = (x + 6)(x - 3) are (-6, 0) and (3, 0). By analyzing the provided options, we can determine that the correct answer is C. (-6, 0). This point is where the graph of the function intersects the x-axis, making it a crucial feature of the quadratic function. Understanding how to find x-intercepts, especially from the factored form, is essential for solving quadratic equations and graphing parabolas. This process not only helps in identifying the roots of the equation but also provides valuable insights into the function's behavior and graphical representation. Mastering these concepts is crucial for success in algebra and beyond.
By setting each factor of the quadratic function equal to zero, we efficiently found the x-intercepts. This method highlights the power of the factored form in revealing key characteristics of the quadratic function. Remember, the x-intercepts are the points where the function's graph crosses the x-axis, and their determination is a fundamental step in understanding the function's behavior. Always double-check your calculations and ensure that the points you identify indeed make the function's value zero. This careful approach will lead to accurate solutions and a deeper understanding of quadratic functions.
