Graphing The Parabola Y=(x+1)^2-2 A Step-by-Step Guide

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In this comprehensive guide, we will explore the process of graphing the parabola defined by the equation y = (x + 1)^2 - 2. Parabolas, which are U-shaped curves, play a crucial role in various fields, including physics, engineering, and mathematics. Understanding how to graph them is essential for solving problems and visualizing quadratic functions. Our focus will be on plotting five key points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. This method provides a clear and accurate representation of the parabola's shape and position on the coordinate plane.

Understanding the Parabola Equation

To effectively graph the parabola, we first need to understand the equation y = (x + 1)^2 - 2. This equation is in the vertex form of a quadratic equation, which is given by:

y = a(x - h)^2 + k

Where:

  • (h, k) represents the vertex of the parabola.
  • a determines the direction and width of the parabola.

In our equation, y = (x + 1)^2 - 2, we can identify the following:

  • a = 1 (Since there is no coefficient explicitly written before the parenthesis, it is implied to be 1).
  • h = -1 (Notice that in the equation (x + 1), it is equivalent to (x - (-1)), so h = -1).
  • k = -2

Thus, the vertex of our parabola is (-1, -2). The value of a being 1 indicates that the parabola opens upwards and has a standard width. Understanding these parameters is crucial for accurately plotting the parabola.

Identifying the Vertex

The vertex is the most critical point when graphing a parabola. It is the point where the parabola changes direction – either the lowest point (minimum) if the parabola opens upwards or the highest point (maximum) if it opens downwards. In the vertex form equation y = a(x - h)^2 + k, the vertex is given by the coordinates (h, k).

For our equation, y = (x + 1)^2 - 2, we determined that h = -1 and k = -2. Therefore, the vertex of the parabola is the point (-1, -2). This point serves as the central reference for plotting the rest of the parabola. Knowing the vertex allows us to easily find other points by choosing x-values to the left and right of it, maintaining symmetry and accuracy in our graph. The vertex not only anchors the parabola on the coordinate plane but also gives us valuable information about its orientation and position.

Determining the Direction and Width

The coefficient a in the vertex form equation y = a(x - h)^2 + k plays a pivotal role in determining the parabola's direction and width. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The magnitude of a also affects the width of the parabola. A larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider one.

In our equation, y = (x + 1)^2 - 2, the coefficient a = 1. Since 1 is positive, we know that the parabola opens upwards. The fact that a is 1 (neither a large nor a small number) indicates that the parabola has a standard width, neither particularly narrow nor wide. This understanding of a helps us visualize the general shape of the parabola before we even plot any points, making the graphing process more intuitive and accurate.

Plotting Points on the Parabola

To accurately graph the parabola y = (x + 1)^2 - 2, we will plot five points: the vertex, two points to the left of the vertex, and two points to the right of the vertex. This approach ensures a balanced and precise representation of the parabola's curve.

Step-by-Step Guide to Plotting Points

  1. Identify the Vertex: We have already determined that the vertex is (-1, -2). Plot this point on the coordinate plane. The vertex is the central point around which the parabola is symmetric, making it our primary reference point.

  2. Choose x-values to the Left of the Vertex: Select two x-values that are to the left of the vertex's x-coordinate (-1). Let's choose x = -2 and x = -3. These points will help us define the curve of the parabola on the left side of the vertex.

  3. Calculate the Corresponding y-values: Substitute the chosen x-values into the equation y = (x + 1)^2 - 2 to find the corresponding y-values.

    • For x = -2: y = (-2 + 1)^2 - 2 = (-1)^2 - 2 = 1 - 2 = -1 So, the point is (-2, -1).
    • For x = -3: y = (-3 + 1)^2 - 2 = (-2)^2 - 2 = 4 - 2 = 2 So, the point is (-3, 2).

  4. Choose x-values to the Right of the Vertex: Select two x-values that are to the right of the vertex's x-coordinate (-1). Let's choose x = 0 and x = 1. These points will mirror the curve of the parabola on the right side of the vertex.

  5. Calculate the Corresponding y-values: Substitute the chosen x-values into the equation y = (x + 1)^2 - 2 to find the corresponding y-values.

    • For x = 0: y = (0 + 1)^2 - 2 = (1)^2 - 2 = 1 - 2 = -1 So, the point is (0, -1).
    • For x = 1: y = (1 + 1)^2 - 2 = (2)^2 - 2 = 4 - 2 = 2 So, the point is (1, 2).

  6. Plot the Points: Plot the calculated points (-2, -1), (-3, 2), (0, -1), and (1, 2) on the coordinate plane.

Creating the Parabola Graph

With the five key points plotted—the vertex (-1, -2), two points to the left (-2, -1) and (-3, 2), and two points to the right (0, -1) and (1, 2)—we can now sketch the parabola. The points provide a clear outline of the curve, making it easier to draw an accurate representation.

  1. Connect the Points: Begin by drawing a smooth, U-shaped curve that passes through all the plotted points. The vertex should be the lowest point of the curve since our parabola opens upwards (as determined by the positive coefficient a). The curve should be symmetrical around the vertical line that passes through the vertex (the axis of symmetry).

  2. Extend the Curve: Extend the curve beyond the plotted points to indicate that the parabola continues infinitely in both upward directions. The extensions should maintain the smooth, U-shaped form, ensuring the graph accurately represents the parabolic function.

  3. Check for Symmetry: Ensure that the graph is symmetrical around the axis of symmetry, which is the vertical line x = -1 (the x-coordinate of the vertex). This symmetry is a fundamental characteristic of parabolas and should be visually evident in your graph.

  4. Review the Shape: Review the overall shape of the parabola to confirm that it aligns with our earlier analysis. It should open upwards, have a standard width, and have its vertex at (-1, -2). Any deviations from these characteristics may indicate a plotting error.

Key Features of the Graph

Analyzing the graph of the parabola y = (x + 1)^2 - 2 reveals several key features that are important for understanding the function's behavior and characteristics.

Vertex and Axis of Symmetry

The vertex, as we've established, is the point (-1, -2). It represents the minimum value of the function since the parabola opens upwards. The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. For this parabola, the axis of symmetry is the line x = -1. The vertex and axis of symmetry are fundamental in understanding the parabola's orientation and symmetry.

Intercepts

  • y-intercept: The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, we set x = 0 in the equation: y = (0 + 1)^2 - 2 = 1 - 2 = -1 So, the y-intercept is (0, -1).
  • x-intercepts: The x-intercepts are the points where the parabola intersects the x-axis. To find the x-intercepts, we set y = 0 in the equation and solve for x: 0 = (x + 1)^2 - 2 (x + 1)^2 = 2 x + 1 = ±√2 x = -1 ± √2 Thus, the x-intercepts are (-1 + √2, 0) and (-1 - √2, 0). These points provide additional context to the parabola's position on the coordinate plane.

Domain and Range

  • Domain: The domain of a quadratic function is all real numbers because there are no restrictions on the x-values that can be inputted into the equation. In interval notation, the domain is (-∞, ∞).
  • Range: The range is the set of all possible y-values. Since the parabola opens upwards and the vertex is at (-1, -2), the minimum y-value is -2. Therefore, the range is all y-values greater than or equal to -2. In interval notation, the range is [-2, ∞). The domain and range provide a comprehensive view of the function's scope and limitations.

Conclusion

Graphing the parabola y = (x + 1)^2 - 2 involves several key steps: identifying the vertex, determining the direction and width, plotting five strategic points (the vertex, two points to the left, and two points to the right), and connecting these points to form the parabolic curve. By understanding the vertex form of a quadratic equation and the significance of the coefficients, we can accurately graph parabolas and analyze their key features, such as the vertex, axis of symmetry, intercepts, domain, and range. This process not only enhances our understanding of quadratic functions but also provides a valuable tool for solving various mathematical and real-world problems involving parabolic relationships.