Finding The X-Intercepts Of C(x) = -(x-6)(x+7)

Understanding the behavior of functions is a cornerstone of mathematics, and a critical aspect of this understanding lies in identifying the x-intercepts. The x-intercepts are the points where the graph of a function crosses the x-axis, representing the values of x for which the function's output, c(x) in this case, equals zero. This article delves into a detailed exploration of how to determine the x-intercepts of the function c(x) = -(x-6)(x+7) using its graphical representation. We will not only pinpoint the correct locations of these intercepts but also provide a comprehensive explanation of the underlying principles. This will empower you to confidently tackle similar problems and gain a deeper understanding of functions and their graphs. Let's embark on this mathematical journey and unlock the secrets hidden within the equation.

Understanding X-Intercepts

Before diving into the specifics of the function c(x) = -(x-6)(x+7), it's crucial to solidify our understanding of x-intercepts. In essence, the x-intercepts are the points where the graph of a function intersects the x-axis. At these points, the y-coordinate (or the function's value, c(x) in our case) is always zero. Therefore, to find the x-intercepts, we need to solve the equation c(x) = 0. Geometrically, these intercepts represent the real roots or solutions of the equation. The number of x-intercepts a function has can provide valuable insights into its behavior and characteristics. For instance, a quadratic function (a function with the highest power of x being 2) can have up to two x-intercepts, corresponding to the two possible real roots of the quadratic equation. Understanding this fundamental concept is paramount to analyzing and interpreting graphs of functions. It allows us to connect the algebraic representation of a function with its visual representation, providing a holistic understanding of its behavior. In the context of real-world applications, x-intercepts can represent critical points, such as break-even points in business models or equilibrium points in physical systems. Therefore, mastering the skill of finding x-intercepts is not only essential for academic success but also for practical problem-solving in various fields.

Analyzing the Function c(x) = -(x-6)(x+7)

The function c(x) = -(x-6)(x+7) is a quadratic function in factored form. This form provides us with a direct pathway to finding the x-intercepts. Each factor corresponds to a potential x-intercept. Setting c(x) equal to zero, we get: 0 = -(x-6)(x+7). A product of factors equals zero if and only if at least one of the factors is zero. Therefore, we can set each factor individually equal to zero and solve for x. This gives us two equations: x - 6 = 0 and x + 7 = 0. Solving the first equation, x - 6 = 0, we add 6 to both sides, resulting in x = 6. This indicates that there is an x-intercept at the point where x is 6. Solving the second equation, x + 7 = 0, we subtract 7 from both sides, resulting in x = -7. This indicates that there is another x-intercept at the point where x is -7. Therefore, the function c(x) = -(x-6)(x+7) has two x-intercepts, corresponding to the values x = 6 and x = -7. These are the points where the parabola represented by the function crosses the x-axis. The negative sign in front of the factored form indicates that the parabola opens downwards. This knowledge, combined with the location of the x-intercepts, helps us sketch a general shape of the graph of the function. The vertex of the parabola, which represents the maximum point of the function, will lie midway between the x-intercepts.

Determining the X-Intercept Locations

Now that we have identified the x-values where the function c(x) = -(x-6)(x+7) intersects the x-axis, we can express these points as coordinates. Recall that x-intercepts occur where the function's value, c(x), is zero. Therefore, the y-coordinate of each x-intercept is always 0. We found that the function has x-intercepts at x = 6 and x = -7. To represent these as coordinate pairs, we pair each x-value with a y-value of 0. This gives us the points (6, 0) and (-7, 0). These are the precise locations where the graph of the function c(x) = -(x-6)(x+7) crosses the x-axis. Therefore, the correct answer to the question is that the x-intercepts are located at (6, 0) and (-7, 0). The options provided include: A. (-6, 0) and (7, 0), B. (-6, 0) and (-7, 0), C. (6, 0) and (7, 0). Based on our analysis, the correct answer is neither A, B, nor C. We've pinpointed the correct x-intercepts through algebraic manipulation, setting each factor of the function to zero and solving for x. This methodical approach ensures accuracy and a thorough understanding of the function's behavior. By correctly identifying the x-intercepts, we gain valuable insights into the function's graph, its roots, and its overall characteristics. This ability is crucial for further analysis and applications of the function in various mathematical contexts.

Connecting the Algebra to the Graph

The power of mathematics lies in its ability to connect abstract concepts with visual representations. In the case of the function c(x) = -(x-6)(x+7), we've used algebra to determine the x-intercepts, and now we can connect this to the graph of the function. Visualizing the graph of a quadratic function provides a deeper understanding of its properties and behavior. A quadratic function in the form c(x) = a(x-h)(x-k) represents a parabola, where h and k are the x-intercepts. In our function, c(x) = -(x-6)(x+7), we identified the x-intercepts as (6, 0) and (-7, 0). This means the parabola crosses the x-axis at these two points. The negative sign in front of the factored form, which is the "-" in -(x-6)(x+7), indicates that the parabola opens downwards. This is because the coefficient of the x² term when the function is expanded is negative. Knowing the x-intercepts and the direction the parabola opens allows us to sketch a basic graph. We can imagine a parabola that intersects the x-axis at x = 6 and x = -7 and opens downwards. The vertex of the parabola, which is the highest point in this case, will lie on the axis of symmetry, which is the vertical line that passes midway between the two x-intercepts. The axis of symmetry can be found by averaging the x-values of the x-intercepts: (6 + (-7))/2 = -0.5. Therefore, the x-coordinate of the vertex is -0.5. To find the y-coordinate of the vertex, we substitute x = -0.5 into the function c(x): c(-0.5) = -(-0.5-6)(-0.5+7) = -(-6.5)(6.5) = 42.25. So, the vertex is at (-0.5, 42.25). By combining the x-intercepts and the vertex, we can create a more accurate sketch of the graph, further solidifying our understanding of the function.

Conclusion

In this detailed exploration, we have successfully determined the locations of the x-intercepts of the function c(x) = -(x-6)(x+7). By setting the function equal to zero and solving for x, we identified the x-intercepts as (6, 0) and (-7, 0). This process demonstrates the crucial connection between the algebraic representation of a function and its graphical behavior. Understanding x-intercepts is fundamental in analyzing functions, as they represent the points where the graph intersects the x-axis and provide valuable information about the function's roots and behavior. We further connected the algebraic solution to the graph of the function, visualizing a downward-opening parabola intersecting the x-axis at the calculated points. We also determined the vertex of the parabola, which provided a complete understanding of its shape and position. The ability to find x-intercepts and interpret them graphically is a powerful tool in mathematics, allowing us to solve problems and gain deeper insights into the behavior of functions. This understanding extends beyond the classroom, finding applications in various fields where mathematical modeling is used. By mastering these concepts, you can confidently analyze and interpret functions, making informed decisions and solving real-world problems.