Analyzing F(x) = (x-1)(x+7) Graph And Identifying True Statements

This article delves into the analysis of the quadratic function f(x) = (x-1)(x+7), focusing on key characteristics that can be derived from its graphical representation. We will explore how to determine the vertex of the parabola, understand the implications of the leading coefficient, and identify crucial points such as the x-intercepts and y-intercept. By carefully examining these aspects, we can accurately assess the truthfulness of various statements about the function. Understanding the properties of quadratic functions is essential in mathematics, as they model various real-world scenarios, from projectile motion to optimization problems. Therefore, a thorough analysis of f(x) = (x-1)(x+7) provides valuable insights into the behavior of parabolas and their applications. This exploration will not only help in answering specific questions about this function but also enhance our overall understanding of quadratic equations and their graphical representations. This article aims to break down the process of analyzing a quadratic function step by step, making it accessible and informative for anyone interested in deepening their mathematical knowledge. The graphical representation of a function is a powerful tool for understanding its behavior. In the case of quadratic functions, the graph is a parabola, a U-shaped curve that opens either upwards or downwards. The key features of a parabola, such as its vertex, axis of symmetry, and intercepts, provide valuable information about the function itself. By analyzing the given graph of f(x) = (x-1)(x+7), we can determine the coordinates of the vertex, which represents the maximum or minimum point of the function. The vertex is crucial because it helps us understand the range of the function and its overall shape. Additionally, the x-intercepts, where the graph crosses the x-axis, correspond to the roots or solutions of the quadratic equation f(x) = 0. These intercepts are essential for solving various problems related to quadratic functions. The y-intercept, where the graph crosses the y-axis, gives us the value of the function when x = 0. By carefully examining these points on the graph, we can piece together a comprehensive understanding of the function's behavior and its mathematical properties. This visual analysis is a fundamental step in solving problems involving quadratic functions and understanding their real-world applications.

Determining the Vertex of the Function

The vertex of a parabola, which represents the quadratic function, holds significant importance as it indicates the function's minimum or maximum value. To accurately determine the vertex, we can employ several methods. One approach is to convert the given function, f(x) = (x-1)(x+7), into its standard form, f(x) = ax² + bx + c. By expanding the factored form, we get f(x) = x² + 6x - 7. The x-coordinate of the vertex can then be found using the formula x = -b / 2a. In this case, a = 1 and b = 6, so the x-coordinate of the vertex is x = -6 / (2 * 1) = -3. To find the y-coordinate, we substitute this x-value back into the function: f(-3) = (-3)² + 6(-3) - 7 = 9 - 18 - 7 = -16. Therefore, the vertex of the function is located at the point (-3, -16). Another method to find the vertex involves recognizing that the x-coordinate of the vertex lies exactly midway between the x-intercepts of the parabola. The x-intercepts are the points where the graph crosses the x-axis, which occur when f(x) = 0. From the given function f(x) = (x-1)(x+7), we can see that the x-intercepts are x = 1 and x = -7. The midpoint between these two x-intercepts is (-7 + 1) / 2 = -3, which confirms the x-coordinate of the vertex. By substituting x = -3 into the function, we again find that the y-coordinate is -16, thus verifying the vertex coordinates. Understanding the vertex is crucial because it allows us to determine the minimum or maximum value of the function. In this case, since the coefficient of the x² term is positive (a = 1), the parabola opens upwards, and the vertex represents the minimum point of the function. This means that the function f(x) = (x-1)(x+7) has a minimum value of -16, which occurs at x = -3. The vertex also helps us understand the symmetry of the parabola. The vertical line passing through the vertex, known as the axis of symmetry, divides the parabola into two mirror images. For the function f(x) = (x-1)(x+7), the axis of symmetry is the vertical line x = -3. This symmetry is a fundamental property of quadratic functions and is useful in various applications. By accurately determining the vertex, we gain valuable insights into the behavior and characteristics of the quadratic function, which aids in solving related problems and understanding real-world applications.

Analyzing the Graph and Identifying True Statements

When analyzing the graph of the function f(x) = (x-1)(x+7), several key features stand out that allow us to evaluate the truthfulness of different statements. The graph is a parabola, and its shape and position are determined by the function's equation. From the factored form f(x) = (x-1)(x+7), we can identify the x-intercepts as x = 1 and x = -7, as these are the values of x that make f(x) = 0. The x-intercepts are crucial points because they provide information about the roots of the quadratic equation. Additionally, the y-intercept, which is the point where the graph crosses the y-axis, can be found by setting x = 0 in the function. In this case, f(0) = (0-1)(0+7) = -7, so the y-intercept is at the point (0, -7). Another significant feature of the graph is its vertex, which we determined earlier to be at (-3, -16). The vertex represents the minimum point of the function since the parabola opens upwards (the coefficient of the x² term is positive). This means that the function has a minimum value of -16, and it occurs at x = -3. The vertex is also the point of symmetry for the parabola, and the vertical line passing through the vertex is the axis of symmetry. The axis of symmetry for f(x) = (x-1)(x+7) is the line x = -3. By examining the graph, we can also determine the intervals where the function is increasing or decreasing. Since the parabola opens upwards, the function is decreasing to the left of the vertex (for x < -3) and increasing to the right of the vertex (for x > -3). This information is valuable in understanding the overall behavior of the function. When evaluating statements about the function, it is essential to consider these key features. For instance, if a statement claims that the vertex is at a different point, we can immediately determine its falsity based on our calculated vertex coordinates. Similarly, statements about the x-intercepts, y-intercept, or the function's increasing/decreasing intervals can be verified or refuted by comparing them with the actual graph and the function's properties. In summary, a thorough analysis of the graph of f(x) = (x-1)(x+7) involves identifying the x-intercepts, y-intercept, vertex, axis of symmetry, and the intervals where the function is increasing or decreasing. These features provide a comprehensive understanding of the function's behavior and enable us to accurately assess the truthfulness of various statements about it. This methodical approach ensures a clear and precise analysis, which is essential in mathematical problem-solving.

Conclusion: Key Takeaways from Analyzing f(x) = (x-1)(x+7)

In conclusion, analyzing the graph of the quadratic function f(x) = (x-1)(x+7) provides a wealth of information about its behavior and properties. By systematically examining key features such as the vertex, intercepts, and axis of symmetry, we can gain a comprehensive understanding of the function. The vertex, which we determined to be at (-3, -16), represents the minimum point of the parabola and is crucial for identifying the function's range. The x-intercepts, found at x = 1 and x = -7, are the roots of the equation f(x) = 0 and provide valuable insights into the function's solutions. The y-intercept, located at (0, -7), gives the value of the function when x = 0. The axis of symmetry, the vertical line x = -3, divides the parabola into two symmetrical halves, highlighting the function's balanced nature. Understanding these key features allows us to accurately evaluate statements about the function and make informed conclusions. For example, if a statement incorrectly identifies the vertex or x-intercepts, we can confidently refute it based on our analysis. Similarly, we can determine the intervals where the function is increasing or decreasing by considering the vertex and the parabola's orientation. The process of analyzing quadratic functions and their graphs is fundamental in mathematics and has numerous applications in various fields. Quadratic functions model many real-world phenomena, such as projectile motion, optimization problems, and curve fitting. Therefore, mastering the techniques for analyzing these functions is essential for problem-solving and mathematical proficiency. This article has demonstrated a step-by-step approach to analyzing the graph of f(x) = (x-1)(x+7), emphasizing the importance of identifying key features and using them to make informed conclusions. By applying these methods, anyone can confidently analyze quadratic functions and their graphical representations, enhancing their mathematical skills and understanding. Ultimately, the ability to analyze graphs and functions is a powerful tool in mathematics, enabling us to solve complex problems and gain deeper insights into the world around us. The insights gained from this analysis are not only applicable to this specific function but also provide a framework for understanding other quadratic functions and their properties. This foundational knowledge is crucial for further studies in mathematics and related fields.