Graphing F(x) = (x + 1)(x - 5) A Step-by-Step Guide

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#include #include #include

using namespace std;

/**

  • @brief Structure to represent a point in 2D space. */ struct Point { double x; double y; };

/**

  • @brief Function to find the x-intercepts of the quadratic function f(x) = (x + 1)(x - 5).
  • The x-intercepts are the points where the function crosses the x-axis, i.e., where f(x) = 0.
  • For the given function, the x-intercepts are the solutions to the equation (x + 1)(x - 5) = 0.
  • @return A vector of Point structures representing the x-intercepts. */ vector findXIntercepts() { vector intercepts; intercepts.push_back({-1, 0}); // x + 1 = 0 => x = -1 intercepts.push_back({5, 0}); // x - 5 = 0 => x = 5 return intercepts; }

/**

  • @brief Function to find the midpoint between two points.
  • The midpoint between two points (x1, y1) and (x2, y2) is given by ((x1 + x2) / 2, (y1 + y2) / 2).
  • In this case, we find the midpoint between the x-intercepts to determine the axis of symmetry
  • of the parabola.
  • @param p1 The first point.
  • @param p2 The second point.
  • @return A Point structure representing the midpoint. */ Point findMidpoint(const Point& p1, const Point& p2) { return {(p1.x + p2.x) / 2, (p1.y + p2.y) / 2}; }

/**

  • @brief Function to find the vertex of the parabola.
  • The vertex of the parabola is the point where the function reaches its minimum (or maximum) value.
  • For a quadratic function in the form f(x) = a(x - h)^2 + k, the vertex is at the point (h, k).
  • In this case, we first find the x-coordinate of the vertex, which is the midpoint between the
  • x-intercepts. Then, we substitute this x-coordinate into the function to find the y-coordinate
  • of the vertex.
  • @param midpointX The x-coordinate of the midpoint between the x-intercepts.
  • @return A Point structure representing the vertex of the parabola. */ Point findVertex(double midpointX) { // f(x) = (x + 1)(x - 5) = x^2 - 4x - 5 double vertexY = (midpointX + 1) * (midpointX - 5); return {midpointX, vertexY}; }

/**

  • @brief Function to find additional points on the graph of the function.

  • To get a better understanding of the shape of the graph, we can find additional points by

  • substituting different values of x into the function. Here, we find the y-intercept (the point

  • where the graph crosses the y-axis) and two additional points on either side of the vertex.

  • @return A vector of Point structures representing additional points on the graph. */ vector findAdditionalPoints() { vector points; // y-intercept (x = 0) points.push_back({0, (0 + 1) * (0 - 5)}); // f(0) = -5

    // Points around the vertex (x = 2): x = 1 and x = 3 points.push_back({1, (1 + 1) * (1 - 5)}); // f(1) = -8 points.push_back({3, (3 + 1) * (3 - 5)}); // f(3) = -8

    return points; }

/**

  • @brief Function to display the points in a formatted way.
  • @param points A vector of Point structures to be displayed.
  • @param title A descriptive title for the points being displayed. */ void displayPoints(const vector& points, const string& title) cout << title << endl; for (const auto& point points) { cout << "(" << point.x << ", " << point.y << ") "; cout << endl << endl; }

/**

  • @brief Main function to demonstrate the process of graphing the function f(x) = (x + 1)(x - 5).

  • This function calls other functions to:

    1. Find the x-intercepts.
    1. Find the midpoint between the x-intercepts.
    1. Find the vertex of the parabola.
    1. Find additional points on the graph.
    1. Display all the points.

  • @return 0 if the program executes successfully. */ int main() { // 1. Identify the x-intercepts vector xIntercepts = findXIntercepts(); displayPoints(xIntercepts, "X-Intercepts:");

    // 2. Find the midpoint between the intercepts Point midpoint = findMidpoint(xIntercepts[0], xIntercepts[1]); cout << "Midpoint between the intercepts: (" << midpoint.x << ", " << midpoint.y << ")" << endl << endl;

    // 3. Determine the vertex Point vertex = findVertex(midpoint.x); cout << "Vertex: (" << vertex.x << ", " << vertex.y << ")" << endl << endl;

    // 4. Find additional points vector additionalPoints = findAdditionalPoints(); displayPoints(additionalPoints, "Additional Points:");

    cout << "To graph the function, plot these points on a coordinate plane and draw a smooth curve (parabola) through them." << endl;

    return 0; }

Understanding the Function f(x) = (x + 1)(x - 5)

When graphing the function f(x) = (x + 1)(x - 5), a systematic approach is crucial for accuracy and understanding. This guide will walk you through the essential steps, ensuring you can confidently plot this quadratic function. We'll start by identifying the x-intercepts, then move on to finding the midpoint between these intercepts, which helps us locate the axis of symmetry and the vertex. Finally, we'll discuss how to determine additional points to create a comprehensive and accurate graph. Each step is designed to provide a clear understanding of the function's behavior and characteristics.

1. Identifying the x-intercepts

The x-intercepts are the points where the graph of the function intersects the x-axis. These points are crucial because they provide immediate insights into the function's roots or zeros. To find the x-intercepts, we set f(x) = 0 and solve for x. For the function f(x) = (x + 1)(x - 5), this means solving the equation (x + 1)(x - 5) = 0. This equation is already factored, making it straightforward to find the solutions. We set each factor equal to zero:

  • x + 1 = 0 gives us x = -1, which corresponds to the point (-1, 0).
  • x - 5 = 0 gives us x = 5, which corresponds to the point (5, 0).

Therefore, the x-intercepts are (-1, 0) and (5, 0). These points are the foundation of our graph, acting as anchors that help define the parabola's shape and position. Accurately plotting these intercepts is the first critical step in graphing the function. Understanding x-intercepts is not just about finding points; it's about understanding where the function's value is zero, a fundamental concept in algebra and calculus. This step allows us to visualize where the parabola crosses the x-axis, giving us a clear starting point for sketching the curve. Moreover, the distance between the x-intercepts gives us an idea about the spread of the parabola along the x-axis. In this case, the intercepts are 6 units apart, indicating a moderately wide parabola. The x-intercepts also play a key role in determining the axis of symmetry, which we will discuss in the next step. Without accurately identifying these points, the rest of the graph will likely be incorrect, highlighting the importance of this initial step. The x-intercepts are more than just points; they are vital clues to the behavior and characteristics of the quadratic function. Correctly identifying and plotting these intercepts ensures a solid foundation for the rest of the graphing process. They are the first key pieces in understanding the bigger picture of the function's graphical representation.

2. Finding the Midpoint Between the Intercepts

Having identified the x-intercepts, the next crucial step is to find the midpoint between them. This midpoint is significant because it lies on the axis of symmetry of the parabola. The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. Finding the midpoint allows us to pinpoint this axis, which is essential for accurately graphing the function. The midpoint formula is given by M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. In our case, the x-intercepts are (-1, 0) and (5, 0). Applying the midpoint formula:

  • M = ((-1 + 5) / 2, (0 + 0) / 2)
  • M = (4 / 2, 0 / 2)
  • M = (2, 0)

Therefore, the midpoint between the intercepts is (2, 0). This point lies on the axis of symmetry, which is the vertical line x = 2. The x-coordinate of the midpoint, x = 2, is particularly important because it is also the x-coordinate of the vertex of the parabola. The vertex is the point where the parabola changes direction, either the minimum or maximum point of the curve. Finding the midpoint thus provides us with the x-coordinate of the vertex, a critical piece of information for graphing the function. Understanding the axis of symmetry is fundamental to grasping the symmetrical nature of parabolas. It ensures that the graph is balanced and accurately reflects the function's behavior on both sides of the vertex. The midpoint also serves as a reference point for plotting additional points on the graph. By choosing points equidistant from the axis of symmetry, we can easily generate symmetrical pairs of points, further aiding in the creation of an accurate graph. The process of finding the midpoint is not just a mathematical calculation; it's a step that reveals a key structural element of the parabola. This understanding helps in visualizing the curve and predicting its shape. Moreover, knowing the axis of symmetry simplifies the task of finding the vertex, which is often the most important point on the graph. In essence, finding the midpoint is a pivotal step in bridging the gap between the algebraic representation of the function and its graphical depiction. It provides a clear, concrete point that anchors the parabola and guides the subsequent steps in graphing.

3. Determining the Vertex

After finding the midpoint, which gave us the x-coordinate of the vertex, the next step is to determine the complete coordinates of the vertex. As we established, the x-coordinate of the vertex is the same as the x-coordinate of the midpoint, which is x = 2. To find the y-coordinate, we substitute this value into the original function, f(x) = (x + 1)(x - 5):

  • f(2) = (2 + 1)(2 - 5)
  • f(2) = (3)(-3)
  • f(2) = -9

Therefore, the vertex of the parabola is at the point (2, -9). The vertex is a critical point because it represents either the minimum or maximum value of the function. In this case, since the coefficient of the x² term in the expanded form of the function (which is x² - 4x - 5) is positive, the parabola opens upwards, and the vertex represents the minimum point. Knowing the vertex allows us to understand the overall shape and position of the parabola. It's the turning point of the graph, and its y-coordinate indicates the lowest value the function attains. This information is invaluable for sketching an accurate graph. The vertex, along with the x-intercepts, provides a skeletal framework for the parabola. By plotting these three points, we can begin to visualize the curve's trajectory and ensure that our graph reflects the function's key characteristics. The vertex also helps in understanding the range of the function. Since the parabola opens upwards and the vertex is at (2, -9), we know that the function's values will always be greater than or equal to -9. This insight is crucial for interpreting the function's behavior and its implications in various contexts. Moreover, the vertex can be used to rewrite the quadratic function in vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex. This form provides a clear representation of the parabola's transformations compared to the basic parabola y = x². In summary, determining the vertex is a pivotal step in graphing the function because it reveals the function's minimum (or maximum) value, provides a crucial reference point for the curve, and aids in understanding the function's range and overall behavior. Accurately finding and plotting the vertex is essential for a comprehensive understanding of the function's graphical representation.

4. Finding Additional Points to Refine the Graph

With the x-intercepts and vertex plotted, we have a foundational understanding of the parabola's shape and position. However, to create a more accurate and detailed graph, it's beneficial to find additional points. These points help to refine the curve and ensure that the graph truly represents the function f(x) = (x + 1)(x - 5). A strategic approach to finding additional points involves choosing x-values on either side of the axis of symmetry (x = 2 in this case). This leverages the symmetry of the parabola, allowing us to find pairs of points that are equidistant from the axis of symmetry and have the same y-value. For example, we can choose x = 0 and x = 4, which are both 2 units away from the axis of symmetry. Let's calculate the corresponding y-values:

  • For x = 0:
    • f(0) = (0 + 1)(0 - 5) = (1)(-5) = -5
    • So, the point is (0, -5).
  • For x = 4:
    • f(4) = (4 + 1)(4 - 5) = (5)(-1) = -5
    • So, the point is (4, -5).

These two points, (0, -5) and (4, -5), confirm the parabola's symmetry and provide additional guidance for sketching the curve. We can also choose other points, such as x = -1 and x = 5 (which we already know are x-intercepts), or x = 1 and x = 3 to get an even better sense of the graph's shape. Let's calculate the y-values for x = 1 and x = 3:

  • For x = 1:
    • f(1) = (1 + 1)(1 - 5) = (2)(-4) = -8
    • So, the point is (1, -8).
  • For x = 3:
    • f(3) = (3 + 1)(3 - 5) = (4)(-2) = -8
    • So, the point is (3, -8).

These points, (1, -8) and (3, -8), are also symmetrical and help to further define the curve's shape. By plotting these additional points, we can create a smooth, accurate representation of the parabola. The more points we plot, the more confident we can be in the accuracy of our graph. These additional points also provide insights into the function's behavior over different intervals. They allow us to see how the function changes as x varies, reinforcing our understanding of its characteristics. In essence, finding additional points is a refinement process that transforms a basic understanding of the parabola into a detailed and accurate graphical representation. This step ensures that our graph not only captures the key features of the function (intercepts, vertex) but also reflects its overall behavior with precision.

Conclusion

Graphing the function f(x) = (x + 1)(x - 5) involves a series of logical steps that build upon each other. We started by identifying the x-intercepts, which provided the foundational points where the parabola crosses the x-axis. Then, we found the midpoint between these intercepts, leading us to the axis of symmetry and the x-coordinate of the vertex. Determining the vertex was a crucial step, as it revealed the minimum point of the parabola and its overall position in the coordinate plane. Finally, we found additional points to refine the graph and ensure an accurate representation of the function's behavior. This systematic approach not only allows us to graph the function accurately but also deepens our understanding of quadratic functions and their graphical properties. Each step provides valuable insights into the function's characteristics, from its roots to its turning point and overall shape. By following these steps, anyone can confidently graph quadratic functions and gain a deeper appreciation for their mathematical properties and real-world applications. The process of graphing is not just about plotting points; it's about understanding the relationship between the algebraic representation of a function and its visual depiction. This understanding is a fundamental skill in mathematics and is essential for various fields, including physics, engineering, and economics. By mastering the steps outlined in this guide, you'll be well-equipped to tackle more complex graphing challenges and to apply your knowledge of functions to a wide range of problems. Remember, practice is key to mastering any mathematical skill. The more you graph quadratic functions, the more intuitive the process will become. You'll start to recognize patterns, anticipate the shape of the graph, and develop a deeper understanding of the interplay between the function's equation and its graphical representation. So, take the time to work through examples, experiment with different functions, and enjoy the process of bringing mathematics to life through graphing.