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

Determining the direction a parabola opens is a fundamental concept in understanding quadratic functions. The direction of a parabola described by a quadratic function is dictated by the sign of its leading coefficient. In this article, we will delve into how to identify the direction of a parabola, specifically focusing on the function g(x) = -1/2x² + x + 0.5. We'll break down the components of the quadratic function, explain the significance of the leading coefficient, and provide a clear explanation of why the correct answer is C. Down.

Decoding Quadratic Functions and Parabolas

Before we tackle the specific function, let's establish a foundation by understanding the basics of quadratic functions and their graphical representations as parabolas. A quadratic function is a polynomial function of degree two, generally expressed in the form:

f(x) = ax² + bx + c

where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a U-shaped curve called a parabola. This curve can open either upwards or downwards, depending on the value of the coefficient a, which is also known as the leading coefficient.

The leading coefficient plays a crucial role in defining the parabola's orientation. 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 is due to the ax² term dominating the function's behavior as |x| becomes large. A positive a means that as x moves away from the vertex, the function values increase positively, hence the upward opening. A negative a means that as x moves away from the vertex, the function values decrease negatively, hence the downward opening.

The constants b and c also influence the parabola's characteristics, but they do not determine the direction of opening. The coefficient b affects the position of the vertex (the turning point of the parabola) along the x-axis, and c represents the y-intercept of the parabola (the point where the parabola intersects the y-axis). Therefore, to determine the direction of the parabola, we primarily focus on the sign of a, the leading coefficient.

Analyzing the Given Function: g(x) = -1/2x² + x + 0.5

Now, let's apply this knowledge to the function given in the question: g(x) = -1/2x² + x + 0.5. Our goal is to identify the direction in which the parabola described by this function opens. To do this, we need to examine the leading coefficient of the function.

In this case, the function is already in the standard quadratic form f(x) = ax² + bx + c. We can directly identify the coefficients:

• a = -1/2
• b = 1
• c = 0.5

As we established earlier, the direction of the parabola is determined by the sign of the leading coefficient a. Here, a is -1/2, which is a negative value. Therefore, according to our understanding of quadratic functions, the parabola opens downwards.

To further illustrate this, consider what happens to the function's value as x moves away from the vertex. Since the coefficient of the x² term is negative, the term -1/2x² will become increasingly negative as |x| increases. This negative contribution dominates the function's behavior for large |x|, causing the function values to decrease, thus resulting in a downward-opening parabola.

We can also visualize this by plotting a few points or using graphing software. You'll notice that the parabola forms a U-shape that opens downwards, confirming our analysis based on the leading coefficient.

Why the Other Options Are Incorrect

To solidify our understanding, let's briefly discuss why the other options provided in the question are incorrect:

• A. Up: A parabola opens upwards when the leading coefficient a is positive. Since our a is negative (-1/2), this option is incorrect.
• B. Right: Parabolas open upwards or downwards, not to the right or left. The direction of opening is along the y-axis, not the x-axis. The concepts of "right" and "left" opening parabolas are typically associated with parabolas defined by equations of the form x = ay² + by + c, where the parabola opens to the right if a is positive and to the left if a is negative. However, in this case, we have a function of the form y = ax² + bx + c, which describes a parabola opening either upwards or downwards.
• D. Left: Similar to option B, parabolas defined by functions of the form y = ax² + bx + c open either upwards or downwards, not to the left or right. Leftward opening parabolas are described by a different form of equation.

Therefore, the only logical answer based on our analysis is that the parabola opens downwards.

Conclusion: The Parabola Opens Downwards

In conclusion, the direction of the parabola described by the function g(x) = -1/2x² + x + 0.5 opens downwards (Option C). This is because the leading coefficient a in the quadratic function is -1/2, which is a negative value. Remember that a negative leading coefficient indicates a downward-opening parabola, while a positive leading coefficient indicates an upward-opening parabola. By understanding this fundamental principle, you can easily determine the direction of any parabola given its quadratic function.

Understanding the relationship between the leading coefficient and the direction of a parabola is a crucial skill in algebra and calculus. This knowledge allows you to quickly visualize the shape and behavior of quadratic functions, which are widely used in various applications, such as physics (projectile motion), engineering (designing parabolic reflectors), and economics (modeling cost curves). Mastering these concepts will not only help you solve mathematical problems but also provide a valuable tool for understanding and modeling real-world phenomena. Always remember to pay close attention to the sign of the leading coefficient when analyzing quadratic functions and parabolas. This simple step can provide significant insights into the function's behavior and graphical representation.