Analyzing The Properties Of The Quadratic Function F(x) = 2x² - X - 6

In the realm of mathematics, quadratic functions hold a prominent position, and understanding their characteristics is crucial for various applications. This article delves into the intricacies of the quadratic function f(x) = 2x² - x - 6, aiming to dissect its key features and shed light on its graphical representation. We will explore the function's domain, range, intercepts, vertex, and axis of symmetry, ultimately revealing the true nature of this mathematical entity.

Dissecting the Domain and Range: Unveiling the Function's Boundaries

The domain of a function encompasses all possible input values (x-values) for which the function yields a valid output. For the quadratic function f(x) = 2x² - x - 6, there are no inherent restrictions on the input values. We can substitute any real number for 'x' and obtain a corresponding output value. Therefore, the domain of this function extends across the entire real number line, represented as (-∞, ∞).

The range, on the other hand, represents the set of all possible output values (y-values) that the function can produce. Unlike the domain, the range of a quadratic function is limited due to the parabolic shape of its graph. The parabola either opens upwards or downwards, depending on the coefficient of the x² term. In our case, the coefficient of x² is 2, which is positive, indicating that the parabola opens upwards. This implies that the function has a minimum value, and the range consists of all y-values greater than or equal to this minimum.

To determine the minimum value, we need to find the vertex of the parabola. The vertex represents the point where the parabola changes direction. For a quadratic function in the standard form of f(x) = ax² + bx + c, the x-coordinate of the vertex is given by -b/(2a). In our function, a = 2 and b = -1, so the x-coordinate of the vertex is -(-1)/(22) = 1/4. Substituting this value back into the function, we get f(1/4) = 2(1/4)² - (1/4) - 6 = -6.125. Therefore, the vertex of the parabola is at the point (1/4, -6.125), and the minimum value of the function is -6.125. Consequently, the range of the function is all real numbers greater than or equal to -6.125, represented as [-6.125, ∞).

In essence, understanding the domain and range of a quadratic function provides crucial insights into the function's behavior and limitations. The domain tells us which input values are permissible, while the range reveals the spectrum of possible output values. For the function f(x) = 2x² - x - 6, the domain encompasses all real numbers, while the range is bounded below by -6.125, reflecting the parabolic nature of the function's graph.

Unveiling Intercepts: Where the Function Meets the Axes

Intercepts are the points where the graph of a function intersects the coordinate axes. These points provide valuable information about the function's behavior and its relationship to the coordinate system. There are two types of intercepts: x-intercepts and y-intercepts.

X-intercepts are the points where the graph intersects the x-axis. At these points, the y-coordinate is zero. To find the x-intercepts of the function f(x) = 2x² - x - 6, we need to solve the equation 2x² - x - 6 = 0. This is a quadratic equation, which we can solve by factoring, completing the square, or using the quadratic formula.

Factoring the quadratic expression, we get (2x + 3)(x - 2) = 0. Setting each factor equal to zero, we find two solutions: 2x + 3 = 0, which gives x = -3/2, and x - 2 = 0, which gives x = 2. Therefore, the x-intercepts of the function are (-3/2, 0) and (2, 0). These points represent where the parabola crosses the x-axis.

Y-intercepts are the points where the graph intersects the y-axis. At these points, the x-coordinate is zero. To find the y-intercept of the function f(x) = 2x² - x - 6, we simply substitute x = 0 into the function: f(0) = 2*(0)² - (0) - 6 = -6. Therefore, the y-intercept of the function is (0, -6). This point represents where the parabola crosses the y-axis.

In summary, intercepts provide critical landmarks on the graph of a function. The x-intercepts of the function f(x) = 2x² - x - 6 are (-3/2, 0) and (2, 0), indicating where the parabola crosses the x-axis. The y-intercept is (0, -6), revealing where the parabola crosses the y-axis. These intercepts, along with the vertex, help us sketch a more accurate representation of the function's graph.

Vertex and Axis of Symmetry: Unveiling the Parabola's Core Structure

The vertex and axis of symmetry are fundamental components of a parabola, providing insights into its shape and orientation. The vertex is the point where the parabola changes direction, representing either the minimum or maximum value of the function. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

As discussed earlier, the x-coordinate of the vertex for a quadratic function in the standard form of f(x) = ax² + bx + c is given by -b/(2a). For the function f(x) = 2x² - x - 6, a = 2 and b = -1, so the x-coordinate of the vertex is -(-1)/(22) = 1/4. To find the y-coordinate of the vertex, we substitute this value back into the function: f(1/4) = 2(1/4)² - (1/4) - 6 = -6.125. Therefore, the vertex of the parabola is at the point (1/4, -6.125).

The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is 1/4, the equation of the axis of symmetry is x = 1/4. This line divides the parabola into two mirror images, ensuring that the function's graph is symmetrical around this line.

The vertex and axis of symmetry provide a framework for understanding the parabola's structure. The vertex (1/4, -6.125) represents the lowest point on the parabola, as the parabola opens upwards. The axis of symmetry x = 1/4 acts as a central dividing line, highlighting the symmetrical nature of the function's graph. These features, combined with the intercepts, offer a comprehensive picture of the quadratic function's behavior.

In conclusion, by carefully analyzing the domain, range, intercepts, vertex, and axis of symmetry of the quadratic function f(x) = 2x² - x - 6, we gain a deep understanding of its properties and graphical representation. This knowledge empowers us to predict the function's behavior, solve related problems, and appreciate the elegance of mathematical concepts. These fundamental aspects of quadratic functions pave the way for further exploration in the world of mathematics.