Analyzing F(x) -6x²-24x-26 Properties Without Graphing

In this article, we will delve into the properties of the quadratic function f(x) = -6x² - 24x - 26 without relying on graphical representations. Our primary focus will be on algebraically determining key characteristics of this function. Specifically, we will explore the concavity of the function, identify its vertex, axis of symmetry, and range. By employing algebraic techniques such as completing the square and analyzing the leading coefficient, we can gain a comprehensive understanding of the function's behavior. This approach not only enhances our analytical skills but also provides a robust method for examining quadratic functions in various mathematical contexts. Understanding these properties is crucial for solving optimization problems, analyzing physical phenomena, and developing a deeper appreciation for the elegance of quadratic functions in mathematics. The methods discussed here provide a foundation for further exploration of more complex functions and their applications in diverse fields.

(a) Determining the Concavity

To determine the concavity of the quadratic function f(x) = -6x² - 24x - 26, we examine the leading coefficient, which is the coefficient of the x² term. In this case, the leading coefficient is -6. The leading coefficient plays a crucial role in determining the shape and direction of the parabola represented by the quadratic function. If the leading coefficient is positive, the parabola opens upwards, indicating a concave up shape. Conversely, if the leading coefficient is negative, the parabola opens downwards, indicating a concave down shape. Since the leading coefficient here is -6, which is negative, the parabola opens downwards. This means that the function f(x) has a concave down shape. A concave down shape implies that the function has a maximum value, which occurs at the vertex of the parabola. Understanding the concavity is fundamental in analyzing quadratic functions because it helps us visualize the overall behavior of the function and predict the existence of maximum or minimum points. Moreover, the concavity provides insights into the rate of change of the function; a concave down function has a decreasing rate of change as x increases, while a concave up function has an increasing rate of change. Therefore, the negative leading coefficient not only tells us about the direction the parabola opens but also gives us valuable information about the function's behavior and its extreme values.

(b) Finding the Vertex and Axis of Symmetry

To find the vertex and axis of symmetry of the quadratic function f(x) = -6x² - 24x - 26, we will use the method of completing the square. Completing the square is an algebraic technique that transforms a quadratic expression into a perfect square trinomial plus a constant. This form allows us to easily identify the vertex of the parabola. First, factor out the leading coefficient, -6, from the x² and x terms: f(x) = -6(x² + 4x) - 26. Next, we need to add and subtract a value inside the parenthesis that completes the square. To find this value, we take half of the coefficient of the x term (which is 4), square it ((4/2)² = 4), and add and subtract it inside the parenthesis: f(x) = -6(x² + 4x + 4 - 4) - 26. Now, rewrite the expression as f(x) = -6((x + 2)² - 4) - 26. Distribute the -6: f(x) = -6(x + 2)² + 24 - 26. Simplify to get the vertex form: f(x) = -6(x + 2)² - 2. The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. In our case, the vertex is (-2, -2). The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. Therefore, the axis of symmetry for this function is x = -2. Understanding the vertex and axis of symmetry provides a clear picture of the parabola's location and orientation in the coordinate plane, which is essential for analyzing its behavior and solving related problems.

(c) Determining the Range

To determine the range of the quadratic function f(x) = -6x² - 24x - 26, we need to consider the concavity of the parabola and the vertex. As we established earlier, the function has a concave down shape because the leading coefficient is negative. This means the parabola opens downwards, and the vertex represents the maximum point of the function. From the vertex form f(x) = -6(x + 2)² - 2, we identified the vertex as (-2, -2). Since the parabola opens downwards, the y-coordinate of the vertex, which is -2, is the maximum value of the function. The range of a function is the set of all possible output values (y-values). For a concave down parabola, the range includes all y-values less than or equal to the y-coordinate of the vertex. Therefore, the range of f(x) is y ≤ -2. In interval notation, this is expressed as (-∞, -2]. Understanding the range is crucial because it tells us the limits of the function's output values. It provides insights into the possible values that the function can attain and is particularly useful in optimization problems where we need to find the maximum or minimum values of a function. The range, combined with the concavity and vertex information, gives a complete picture of the function's behavior and its graphical representation.

(d) Finding the x-intercepts

To find the x-intercepts of the quadratic function f(x) = -6x² - 24x - 26, we need to solve the equation f(x) = 0. This means we need to find the values of x for which the function equals zero. The x-intercepts are the points where the parabola intersects the x-axis. We set the function equal to zero: -6x² - 24x - 26 = 0. To simplify the equation, we can divide all terms by -2: 3x² + 12x + 13 = 0. Now, we can use the quadratic formula to solve for x. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our case, a = 3, b = 12, and c = 13. Plugging these values into the quadratic formula, we get: x = (-12 ± √(12² - 4 * 3 * 13)) / (2 * 3). Simplify the expression under the square root: 12² - 4 * 3 * 13 = 144 - 156 = -12. So, the equation becomes: x = (-12 ± √(-12)) / 6. Since the discriminant (the value under the square root) is negative, there are no real solutions for x. This means that the parabola does not intersect the x-axis. Therefore, the function f(x) has no real x-intercepts. The absence of x-intercepts indicates that the entire parabola lies either above or below the x-axis. In this case, since the parabola opens downwards and the vertex is below the x-axis, the function's graph is entirely below the x-axis. Finding the x-intercepts is a crucial step in analyzing quadratic functions as it provides additional points for graphing and helps in understanding the function's behavior relative to the x-axis.

Conclusion

In summary, we have thoroughly analyzed the quadratic function f(x) = -6x² - 24x - 26 without using graphical methods. We determined that the function has a concave down shape due to the negative leading coefficient. We found the vertex to be at (-2, -2) and the axis of symmetry to be the line x = -2 by completing the square. The range of the function is y ≤ -2, indicating that the maximum value of the function is -2. Furthermore, we determined that the function has no real x-intercepts by using the quadratic formula and observing the negative discriminant. These findings provide a comprehensive understanding of the function's behavior and characteristics. By employing algebraic techniques, we have demonstrated how to analyze quadratic functions effectively, which is crucial for various mathematical and practical applications. This approach not only enhances our problem-solving skills but also provides a solid foundation for exploring more complex mathematical concepts. The ability to analyze functions algebraically is a valuable tool in mathematics and related fields, allowing us to solve problems and make predictions without relying solely on visual representations.