Determining The Direction Of Parabola G(x) = -1/2x^2 + X + 0.5

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Determining the direction a parabola opens is a fundamental concept in understanding quadratic functions. In this comprehensive exploration, we will dissect the function g(x) = -1/2x² + x + 0.5 to reveal whether its graph opens upwards, downwards, leftwards, or rightwards. We will delve into the significance of the leading coefficient and its impact on the parabola's orientation, providing a clear and insightful analysis for students and math enthusiasts alike. This analysis not only answers the specific question but also fortifies your understanding of quadratic functions and their graphical representations.

Decoding the Parabola's Direction

The question at hand is: Which direction does the graph of the parabola described by the function g(x) = -1/2x² + x + 0.5 open towards? The options are A. Up, B. Right, C. Down, and D. Left. To definitively answer this, we need to understand the anatomy of a parabolic equation.

Parabolas, the U-shaped curves, are the graphical representation of quadratic functions. A quadratic function is generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and importantly, a is not equal to zero. The coefficient a, the one multiplying the x² term, plays a pivotal role in determining the parabola's direction.

The Leading Coefficient 'a' and Its Influence

The direction in which a parabola opens is solely determined by the sign of the leading coefficient, a. This is a crucial concept to grasp. If a is positive (a > 0), the parabola opens upwards, resembling a smile. Conversely, if a is negative (a < 0), the parabola opens downwards, resembling a frown. This behavior stems from how the x² term dominates the function's behavior as |x| becomes large. When a is positive, large |x| values result in large positive f(x) values, causing the parabola to open upwards. When a is negative, large |x| values result in large negative f(x) values, causing the parabola to open downwards.

In our given function, g(x) = -1/2x² + x + 0.5, the leading coefficient a is -1/2. This value is decidedly negative. Therefore, based on our understanding of the relationship between a and the parabola's direction, we can confidently conclude that the parabola opens downwards. Options B and D (Right and Left) are incorrect because parabolas, as functions of x, open either upwards or downwards, not sideways. The correct answer is C. Down.

A Deep Dive into Quadratic Functions and Parabolas

To truly master the behavior of parabolas, it's beneficial to explore the broader context of quadratic functions. Quadratic functions are polynomial functions of degree two, meaning the highest power of the variable x is 2. They are ubiquitous in mathematics and the sciences, modeling a wide range of phenomena from projectile motion to the shape of satellite dishes.

The Standard Form and Vertex Form

We've already encountered the standard form of a quadratic function: f(x) = ax² + bx + c. However, there's another valuable form known as the vertex form: f(x) = a(x - h)² + k. The vertex form is particularly insightful because it directly reveals the vertex of the parabola, which is the point where the parabola changes direction. The vertex is located at the point (h, k).

Finding the Vertex

In the standard form, the x-coordinate of the vertex, h, can be found using the formula h = -b / 2a. Once you have h, you can find the y-coordinate, k, by substituting h back into the function: k = f(h). This process allows us to pinpoint the parabola's turning point, which is crucial for sketching the graph and understanding its behavior.

Let's apply this to our function, g(x) = -1/2x² + x + 0.5. Here, a = -1/2 and b = 1. Using the formula, h = -b / 2a = -1 / (2 * -1/2) = 1. Now, we find k: k = g(1) = -1/2(1)² + 1 + 0.5 = -1/2 + 1 + 0.5 = 1. Therefore, the vertex of the parabola is at the point (1, 1).

The Axis of Symmetry

Every parabola possesses a line of symmetry known as the axis of symmetry. This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. The equation of the axis of symmetry is simply x = h, where h is the x-coordinate of the vertex. In our example, the axis of symmetry is the line x = 1.

X-Intercepts and the Quadratic Formula

The x-intercepts of a parabola are the points where the graph intersects the x-axis. These points are also known as the roots or zeros of the quadratic function. To find the x-intercepts, we set f(x) = 0 and solve for x. This often involves using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

The expression b² - 4ac under the square root is called the discriminant. The discriminant provides valuable information about the nature of the roots:

  • If b² - 4ac > 0, the parabola has two distinct real roots (two x-intercepts).
  • If b² - 4ac = 0, the parabola has one real root (the vertex lies on the x-axis).
  • If b² - 4ac < 0, the parabola has no real roots (no x-intercepts).

Let's apply the quadratic formula to our function g(x) = -1/2x² + x + 0.5:

x = (-1 ± √(1² - 4 * -1/2 * 0.5)) / (2 * -1/2)

x = (-1 ± √(1 + 1)) / -1

x = (-1 ± √2) / -1

So, the x-intercepts are x = 1 - √2 and x = 1 + √2. This confirms that our parabola intersects the x-axis at two points.

Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, we set x = 0 and evaluate f(0). In our function, g(0) = -1/2(0)² + 0 + 0.5 = 0.5. Therefore, the y-intercept is the point (0, 0.5).

Graphing the Parabola

With all this information, we can now sketch a fairly accurate graph of g(x) = -1/2x² + x + 0.5. We know it opens downwards, has a vertex at (1, 1), an axis of symmetry at x = 1, x-intercepts at x = 1 - √2 and x = 1 + √2, and a y-intercept at (0, 0.5). Plotting these points and sketching a smooth curve through them gives us a visual representation of the parabola.

Real-World Applications of Parabolas

Parabolas aren't just abstract mathematical concepts; they have numerous real-world applications. For example, the trajectory of a projectile (like a ball thrown in the air) is approximately parabolic, neglecting air resistance. Satellite dishes and reflecting telescopes use parabolic mirrors to focus incoming signals or light to a single point. Suspension bridges often utilize parabolic cables to distribute weight evenly. Even the design of car headlights incorporates parabolic reflectors to create a focused beam of light. Understanding parabolas allows us to analyze and design these systems effectively.

Conclusion

In conclusion, by analyzing the leading coefficient of the quadratic function g(x) = -1/2x² + x + 0.5, we definitively determined that the parabola opens downwards. This exploration has extended beyond the immediate question, delving into the properties of quadratic functions, including the vertex, axis of symmetry, intercepts, and real-world applications. A thorough understanding of these concepts empowers you to analyze and interpret parabolic relationships in various contexts. Mastering the behavior of parabolas is a significant step in your mathematical journey, opening doors to more advanced concepts and applications in mathematics, science, and engineering.

By understanding these fundamental principles, we can confidently analyze and interpret the behavior of parabolas in various contexts. The negative leading coefficient in g(x) = -1/2x² + x + 0.5 serves as a clear indicator that the parabola opens downwards, a key characteristic that shapes its graphical representation and real-world applications.