Bounded By A Quadratic Function Finding The Set Of Values

In the realm of mathematics, quadratic functions hold a prominent position, known for their parabolic curves and versatile applications. This article delves into the intricacies of determining the set of values less than or equal to those represented by a specific quadratic function. We will embark on a step-by-step exploration, starting with the fundamental concepts of quadratic functions and culminating in the identification of the desired set of values.

Understanding Quadratic Functions

At its core, a quadratic function is a polynomial function of degree two, expressed in the general form:

$$f(x) = ax^2 + bx + c$$

where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola's orientation, whether it opens upwards or downwards, is dictated by the sign of the coefficient a. If a is positive, the parabola opens upwards, resembling a smile, while a negative a causes it to open downwards, resembling a frown.

Key features of a parabola include its vertex, the point where the parabola changes direction, and its axis of symmetry, a vertical line passing through the vertex that divides the parabola into two symmetrical halves. The vertex plays a crucial role in determining the minimum or maximum value of the quadratic function. For a parabola opening upwards, the vertex represents the minimum point, while for a parabola opening downwards, it represents the maximum point. The x-coordinate of the vertex is given by -b/2a, and the y-coordinate can be found by substituting this value back into the quadratic function.

Identifying the Vertex and a Point on the Parabola

Our quest begins with a quadratic function characterized by its vertex at the point (-10, -25). This tells us that the parabola either reaches its minimum value of -25 at x = -10 (if it opens upwards) or its maximum value of -25 at x = -10 (if it opens downwards). Furthermore, we are given that the parabola contains the point (-16, -17 4/5), which provides us with additional information about the parabola's shape and position.

The vertex form of a quadratic function proves to be invaluable in this scenario. It expresses the function as:

$$f(x) = a(x - h)^2 + k$$

where (h, k) represents the vertex of the parabola. In our case, the vertex is (-10, -25), so we can substitute these values into the vertex form:

$$f(x) = a(x + 10)^2 - 25$$

To determine the specific quadratic function, we need to find the value of the coefficient a. This is where the point (-16, -17 4/5) comes into play. Since this point lies on the parabola, its coordinates must satisfy the equation of the function. Substituting x = -16 and f(x) = -17 4/5 into the equation, we get:

$$-17 \frac{4}{5} = a(-16 + 10)^2 - 25$$

Simplifying this equation, we have:

$$-17.8 = 36a - 25$$

Adding 25 to both sides:

$$7.2 = 36a$$

Dividing both sides by 36:

$$a = \frac{7.2}{36} = \frac{1}{5}$$

Therefore, the quadratic function we are dealing with is:

$$f(x) = \frac{1}{5}(x + 10)^2 - 25$$

Determining the Set of Values Less Than or Equal to the Quadratic Function

Now that we have the specific quadratic function, our objective is to identify the set of all y-values that are less than or equal to the values produced by this function. In other words, we seek the region below the parabola defined by the function.

Since the coefficient a (1/5) is positive, the parabola opens upwards, indicating that the vertex represents the minimum point of the function. The minimum y-value is -25, which occurs at the vertex (-10, -25). As we move away from the vertex along the x-axis, the y-values increase.

Therefore, the set of values less than or equal to those represented by the quadratic function encompasses all y-values that are below the parabola. This can be expressed mathematically as:

$$y \leq \frac{1}{5}(x + 10)^2 - 25$$

This inequality defines the region in the coordinate plane that lies below or on the parabola. It represents all the points whose y-coordinates are less than or equal to the corresponding y-coordinates on the parabola.

Conclusion

In this exploration, we have successfully determined the set of values less than or equal to those represented by a quadratic function with a given vertex and a point. By leveraging the vertex form of a quadratic function and utilizing the information provided by the given point, we were able to identify the specific quadratic function. Subsequently, we analyzed the parabola's orientation and vertex to deduce the set of values bounded by the function.

This process exemplifies the power of mathematical tools in dissecting and understanding the behavior of functions. Quadratic functions, with their parabolic curves, find applications in various fields, from physics to engineering, making their comprehension essential for problem-solving and modeling real-world phenomena. Understanding how to define the bounds of these functions further expands their utility in various applications.