Graph Of X 2 2x 2

Exploring the Graph of x² + 2x - 2: A Comprehensive Guide

The quadratic function represented by the equation x² + 2x - 2 forms a parabola, a U-shaped curve that's fundamental in algebra and calculus. Understanding its graph involves analyzing key features like its vertex, axis of symmetry, x-intercepts, y-intercept, and overall shape. This comprehensive guide will explore these aspects in detail, offering a thorough understanding of the function's visual representation.

Understanding Quadratic Functions

Before diving into the specifics of x² + 2x - 2, let's establish a foundational understanding of quadratic functions in general. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (x in this case) is 2. It's generally expressed in the form:

f(x) = ax² + bx + c

Where 'a', 'b', and 'c' are constants. The value of 'a' determines the parabola's orientation (opens upwards if a > 0, downwards if a < 0) and its width (smaller |a| means wider parabola). 'b' influences the parabola's horizontal position, and 'c' represents the y-intercept.

Finding the Vertex

The vertex is the turning point of the parabola – the lowest point if it opens upwards or the highest point if it opens downwards. The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

For our equation, x² + 2x - 2, a = 1 and b = 2. Therefore, the x-coordinate of the vertex is:

x = -2 / (2 * 1) = -1

To find the y-coordinate, substitute this x-value back into the original equation:

y = (-1)² + 2(-1) - 2 = -3

Therefore, the vertex of the parabola is (-1, -3).

Significance of the Vertex

The vertex holds significant importance. It represents the minimum (or maximum) value of the function. In real-world applications, this could represent a minimum cost, a maximum profit, or the optimal point in various scenarios. The vertex also serves as a crucial point for sketching the graph accurately.

Determining the Axis of Symmetry

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

x = -1

This line acts as a reference point for plotting symmetrical points on the parabola.

Calculating the x-intercepts (Roots)

The x-intercepts are the points where the parabola intersects the x-axis (where y = 0). To find them, we set the equation equal to zero and solve for x:

x² + 2x - 2 = 0

This quadratic equation doesn't factor easily, so we'll use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values from our equation:

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

Therefore, the x-intercepts are approximately x ≈ 0.732 and x ≈ -2.732.

Significance of the x-intercepts

The x-intercepts represent the solutions or roots of the quadratic equation. They indicate where the function's value equals zero. In applied contexts, these points might represent break-even points, equilibrium points, or other significant thresholds.

Finding the y-intercept

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

y = (0)² + 2(0) - 2 = -2

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

Significance of the y-intercept

The y-intercept represents the initial value or starting point of the function. Its value provides a crucial piece of information for understanding the behaviour of the function at its origin.

Sketching the Graph

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

1. Plot the vertex: (-1, -3)
2. Draw the axis of symmetry: x = -1
3. Plot the x-intercepts: Approximately (0.732, 0) and (-2.732, 0)
4. Plot the y-intercept: (0, -2)
5. Plot a few more points: For instance, you could find the y-values for x = 1 and x = -2 to get a better sense of the parabola's curve. Symmetry simplifies this step – the point (1, y) will have the same y-value as the point (-3, y).
6. Connect the points: Draw a smooth, U-shaped curve through the plotted points, ensuring it's symmetrical around the axis of symmetry.

The graph should clearly show a parabola opening upwards, with its lowest point at the vertex (-1, -3).

Analyzing the Graph's Behavior

The graph illustrates several important behavioral aspects of the quadratic function:

• Increasing and Decreasing Intervals: The function is decreasing for x < -1 and increasing for x > -1.
• Range: The range of the function is [-3, ∞), meaning the y-values are always greater than or equal to -3.
• Domain: The domain of the function is (-∞, ∞), meaning the function is defined for all real numbers.

Applications of Quadratic Graphs

Quadratic functions and their graphs have widespread applications across numerous fields:

• Physics: Modeling projectile motion, where the parabola represents the trajectory of an object.
• Engineering: Designing parabolic antennas and reflectors, which focus signals to a single point.
• Business: Determining optimal production levels or pricing strategies to maximize profits.
• Economics: Analyzing supply and demand curves and modeling economic growth.

Further Exploration: Transformations and Variations

The understanding of the base graph of x² + 2x - 2 can be extended to explore transformations and variations:

• Vertical Shifts: Adding or subtracting a constant from the function shifts the graph vertically.
• Horizontal Shifts: Adding or subtracting a constant within the parentheses shifts the graph horizontally.
• Vertical Stretches/Compressions: Multiplying the entire function by a constant stretches or compresses the graph vertically.
• Horizontal Stretches/Compressions: Multiplying the 'x' term by a constant stretches or compresses the graph horizontally.
• Reflections: Multiplying the entire function by -1 reflects the graph across the x-axis.

Mastering the analysis of the graph of x² + 2x - 2 provides a strong foundation for tackling more complex quadratic functions and their applications in various fields. By understanding its key features and behaviors, you're equipped to interpret and utilize quadratic functions effectively. Remember to utilize online graphing tools to visually reinforce your understanding and experiment with different quadratic functions. The key is to practice and observe the patterns in how the constants affect the shape and position of the parabola.