Graphing The Quadratic Function F(x) = (x + 1)(x - 5)

JU07/08/2025 08, 2025 by THE IDEN 54 views

This article provides a step-by-step guide to graphing the quadratic function f(x) = (x + 1)(x - 5). Understanding how to graph quadratic functions is fundamental in algebra and calculus, as it allows us to visualize the behavior of these equations and identify key features such as x-intercepts, the vertex, and the axis of symmetry. We will delve into each step with detailed explanations and examples, ensuring a comprehensive understanding of the process.

1. Identifying the x-intercepts

To begin graphing this quadratic function, we first need to identify the x-intercepts. The x-intercepts are the points where the graph intersects the x-axis, meaning the y-value (or f(x) value) is zero. For a function in the factored form, like our example f(x) = (x + 1)(x - 5), finding the x-intercepts is straightforward. We set f(x) to zero and solve for x:

(x + 1)(x - 5) = 0

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero:

x + 1 = 0
x - 5 = 0

Solving these equations for x, we get:

x = -1
x = 5

Thus, the x-intercepts are the points (-1, 0) and (5, 0). These points are crucial as they give us two fixed locations on the graph. By understanding the significance of x-intercepts, one can better predict the curve's path and its relation to the x-axis. These are also known as the roots or zeros of the quadratic function. They represent the values of x for which the function equals zero, making them foundational in analyzing quadratic equations.

The process of finding x-intercepts is not just a mathematical exercise; it has practical applications in various fields, including physics and engineering, where parabolic trajectories and curves are common. For example, understanding x-intercepts can help determine the landing point of a projectile or the points of equilibrium in a system. Furthermore, the x-intercepts play a vital role in understanding the nature of the quadratic equation's solutions. If a quadratic equation has two distinct real roots, the parabola will intersect the x-axis at two points. If it has one real root (a repeated root), the parabola will touch the x-axis at one point (the vertex). And if it has no real roots, the parabola will not intersect the x-axis at all. The x-intercepts, therefore, offer valuable insights into the solutions of the equation and the overall shape and position of the parabola.

2. Finding the Midpoint Between the Intercepts

Once we have identified the x-intercepts, the next 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 divides the parabola into two symmetrical halves. The x-coordinate of this midpoint will also be the x-coordinate of the vertex, which is either the minimum or maximum point of the parabola.

To find the midpoint between two points on a number line, we average their x-coordinates. In our case, the x-intercepts are -1 and 5. So, the x-coordinate of the midpoint is:

Midpoint_x = (-1 + 5) / 2

Midpoint_x = 4 / 2

Midpoint_x = 2

Therefore, the x-coordinate of the midpoint is 2. This means the axis of symmetry is the vertical line x = 2. The axis of symmetry is a critical feature of a parabola. It is the line about which the parabola is perfectly symmetric. Visualizing the parabola folded along this line would result in both halves perfectly overlapping. This symmetry is a fundamental property of quadratic functions and is useful in sketching the graph and understanding its behavior. The axis of symmetry not only helps in graphing but also provides insights into the equation's solutions and the function's characteristics.

The midpoint, which we've calculated as x = 2, is crucial for finding the vertex of the parabola. The vertex is the point where the parabola changes direction, either reaching a minimum (if the parabola opens upwards) or a maximum (if the parabola opens downwards). The x-coordinate of the vertex is always the same as the x-coordinate of the midpoint between the intercepts (if there are intercepts) or the x-coordinate of the axis of symmetry. Knowing this midpoint allows us to proceed to the next step, which is finding the y-coordinate of the vertex, and subsequently, the vertex itself. This step is vital in completing the graph, as the vertex is a key point that defines the parabola's overall shape and position in the coordinate plane. The concept of the midpoint extends beyond just graphing quadratic functions; it is a foundational concept in coordinate geometry and is applied in various other mathematical and real-world contexts, such as finding the center of a line segment or determining equilibrium points in physics and engineering problems.

3. Finding the Vertex

Having determined the x-coordinate of the midpoint (which is also the x-coordinate of the vertex), the next crucial step is to find the y-coordinate of the vertex. The vertex is the point where the parabola changes direction, representing either the minimum or maximum value of the function. To find the y-coordinate, we substitute the x-coordinate of the vertex (which we found to be 2) back into the original function:

f(x) = (x + 1)(x - 5)

f(2) = (2 + 1)(2 - 5)

f(2) = (3)(-3)

f(2) = -9

Thus, the y-coordinate of the vertex is -9. Therefore, the vertex of the parabola is the point (2, -9). This point is a cornerstone in graphing the quadratic function because it is the turning point of the parabola. If the coefficient of the x² term in the quadratic function is positive, the parabola opens upwards, and the vertex is the minimum point. Conversely, if the coefficient is negative, the parabola opens downwards, and the vertex is the maximum point. In our case, expanding f(x) = (x + 1)(x - 5) gives us f(x) = x² - 4x - 5, where the coefficient of x² is 1 (positive), indicating that the parabola opens upwards and the vertex (2, -9) is the minimum point.

The vertex provides essential information about the function's range and behavior. Knowing the vertex, we can immediately deduce the minimum or maximum value of the function and understand the function's symmetry. The symmetry is about the vertical line passing through the vertex, which we identified earlier as the axis of symmetry (x = 2). This symmetry means that for any point on the parabola, there is a corresponding point at the same height on the other side of the axis of symmetry. The vertex, in conjunction with the axis of symmetry, gives us a clear picture of the parabola's position and orientation in the coordinate plane. Understanding the vertex and its significance is not just limited to graphing; it's also vital in solving optimization problems in calculus, where finding the maximum or minimum value of a function is a common task. The process of finding the vertex by completing the square or using the formula derived from it is a fundamental skill in algebra, enabling us to transform quadratic equations into a form that readily reveals the vertex coordinates.

4. Sketching the Graph

With the x-intercepts and the vertex determined, we can now sketch the graph of the quadratic function f(x) = (x + 1)(x - 5). We have the following key points:

x-intercepts: (-1, 0) and (5, 0)
Vertex: (2, -9)

Plot these points on a coordinate plane. The parabola will pass through the x-intercepts and have its turning point at the vertex. Since the coefficient of the x² term is positive (as seen when expanding the function), the parabola opens upwards, meaning it has a U-shape.

Draw a smooth curve connecting the points, ensuring the curve is symmetrical about the axis of symmetry (x = 2). The graph should extend upwards from the vertex, passing through the x-intercepts. The shape of the graph, a parabola, is a defining characteristic of quadratic functions. Parabolas are U-shaped curves that can open upwards or downwards, depending on the sign of the leading coefficient (the coefficient of the x² term). The symmetry of the parabola around its axis of symmetry is a key feature that simplifies graphing.

Sketching the graph involves not just connecting the points but also understanding the behavior of the function as x approaches positive and negative infinity. Since our parabola opens upwards, as x moves away from the vertex in either direction, f(x) increases without bound. This behavior is a general characteristic of quadratic functions with a positive leading coefficient. The accuracy of the sketch depends on the precision with which we've plotted the key points and our understanding of the parabola's shape and symmetry. While plotting more points can enhance accuracy, the vertex and x-intercepts provide a solid foundation for a reasonably accurate sketch. In many practical applications, a sketch of the graph is sufficient to understand the qualitative behavior of the function, such as its minimum or maximum value and its roots. Graphing calculators and software can provide more precise graphs, but the ability to sketch a graph by hand remains a valuable skill in mathematics, aiding in conceptual understanding and problem-solving. The graph of a quadratic function provides a visual representation of the function's properties, making it easier to analyze its behavior and extract meaningful information.

Conclusion

Graphing the quadratic function f(x) = (x + 1)(x - 5) involves several key steps: identifying the x-intercepts, finding the midpoint between the intercepts, determining the vertex, and sketching the graph. Each step contributes to a comprehensive understanding of the function's behavior and its graphical representation. By following these steps, one can effectively graph any quadratic function and gain insights into its properties and applications.