Graph Of X 2 2x 2

Unveiling the Secrets of the Graph of x² + 2x - 2

The quadratic function, represented by the equation x² + 2x - 2, offers a rich landscape for exploration within the realm of algebra and graphing. Understanding its graph reveals crucial insights into its behavior, properties, and applications. This comprehensive guide delves deep into the intricacies of this quadratic function, covering everything from its basic characteristics to advanced analysis techniques.

Understanding the Quadratic Equation: x² + 2x - 2

Before we delve into the graphical representation, let's understand the fundamental components of the equation itself: x² + 2x - 2. This is a standard quadratic equation in the form ax² + bx + c, where:

• a = 1: This positive coefficient dictates that the parabola opens upwards (a U-shape).
• b = 2: This coefficient influences the parabola's horizontal position and the x-coordinate of its vertex.
• c = -2: This constant term determines the parabola's vertical intercept (y-intercept).

These coefficients play a critical role in shaping the graph's characteristics. We will see how each of these parameters contributes to the overall shape and position of the parabola.

Key Features of the Parabola

To fully grasp the graph, we must identify its key features:

1. Vertex: The Turning Point

The vertex is the lowest point (or highest point for parabolas that open downwards) on the parabola. Finding the vertex is crucial for accurately sketching the graph. For a quadratic equation in the form ax² + bx + c, the x-coordinate of the vertex is given by:

x = -b / 2a

In our case, x = -2 / (2 * 1) = -1.

Substituting this x-value back into the original equation gives us the y-coordinate:

y = (-1)² + 2(-1) - 2 = -3

Therefore, the vertex of the parabola is (-1, -3). This point represents the minimum value of the function.

2. Axis of Symmetry: A Mirror Image

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is simply:

x = -b / 2a

This mirrors the calculation for the x-coordinate of the vertex. In our case, the axis of symmetry is x = -1. Any point on one side of this line has a corresponding point on the other side with the same y-coordinate.

3. x-Intercepts: Where the Parabola Crosses the x-axis

The x-intercepts (also known as roots or zeros) are the points where the parabola intersects the x-axis (where y = 0). To find these, we set the equation to zero and solve for x:

x² + 2x - 2 = 0

This quadratic equation cannot be easily factored, so we use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Substituting our values (a = 1, b = 2, c = -2), we get:

x = [-2 ± √(2² - 4 * 1 * -2)] / (2 * 1) = [-2 ± √12] / 2 = -1 ± √3

Therefore, the x-intercepts are approximately (-2.732, 0) and (0.732, 0).

4. y-Intercept: Where the Parabola Crosses the y-axis

The y-intercept is the point where the parabola intersects the y-axis (where x = 0). To find this, we simply substitute x = 0 into the equation:

y = (0)² + 2(0) - 2 = -2

Thus, the y-intercept is (0, -2).

Sketching the Graph

Now that we have identified the key features – vertex, axis of symmetry, x-intercepts, and y-intercept – we can accurately sketch the graph.

1. Plot the vertex: Mark the point (-1, -3) on the coordinate plane.

2. Draw the axis of symmetry: Draw a vertical line passing through the vertex at x = -1.

3. Plot the intercepts: Mark the x-intercepts approximately at (-2.732, 0) and (0.732, 0), and the y-intercept at (0, -2).

4. Sketch the parabola: Using the symmetry of the parabola, sketch a smooth curve that passes through all the plotted points, ensuring it opens upwards since a = 1.

This graphical representation visually confirms the properties we calculated.

Advanced Analysis: Exploring Further

Beyond the basic sketching, we can perform more advanced analyses to gain deeper understanding:

1. Domain and Range

• Domain: The domain of a quadratic function is all real numbers, as there are no restrictions on the x-values. We can input any real number into the equation and obtain a corresponding y-value. Domain: (-∞, ∞)

• Range: Since the parabola opens upwards and has a minimum point at the vertex (-1, -3), the range consists of all y-values greater than or equal to -3. Range: [-3, ∞)

2. Increasing and Decreasing Intervals

• Increasing: The function is increasing for all x-values greater than the x-coordinate of the vertex (-1). Increasing Interval: (-1, ∞)

• Decreasing: The function is decreasing for all x-values less than the x-coordinate of the vertex (-1). Decreasing Interval: (-∞, -1)

3. Concavity

The parabola opens upwards, indicating that it is concave up. This is directly related to the positive value of the coefficient 'a'.

4. Using Calculus

For a more rigorous analysis, we can employ calculus. The first derivative of the function, f'(x) = 2x + 2, provides information about the slope of the tangent line at any point. Setting f'(x) = 0, we find the critical point x = -1, confirming the x-coordinate of the vertex. The second derivative, f''(x) = 2, is positive, indicating that the function is concave up.

Applications of Quadratic Functions

Quadratic functions are not merely abstract mathematical concepts; they find extensive applications in various real-world scenarios:

• Projectile Motion: The trajectory of a projectile under the influence of gravity can be modeled using a quadratic function.

• Area and Optimization: Quadratic functions are used to determine maximum or minimum areas of shapes, leading to optimization problems in engineering and design.

• Economics and Business: They appear in cost, revenue, and profit models, assisting businesses in making informed decisions.

• Physics and Engineering: Many physical phenomena, like the path of a ball thrown in the air or the shape of a hanging cable, are described by quadratic equations.

• Computer Graphics: Quadratic curves are fundamental to creating smooth and realistic shapes in computer-aided design and animation.

Understanding the graph of x² + 2x - 2 provides a foundational understanding of quadratic functions and their applications. By mastering the techniques presented here, you can confidently analyze and interpret the behavior of this and other quadratic equations, unlocking their potential for practical problem-solving. This detailed exploration moves beyond a simple graph, empowering you to comprehend the underlying mathematical principles and real-world significance of quadratic functions.