Parabolas F(x)=2(x+8)(x-1) And G(x)=-3/4(x+8)(x-1) Common Characteristics

#Parabolas, defined as quadratic functions, exhibit a unique U-shaped curve. This article delves into the commonalities between two specific parabolas: f(x) = 2(x + 8)(x - 1) and g(x) = -3/4(x + 8)(x - 1). By analyzing their equations, we can unveil shared characteristics and differences, providing insights into the behavior of quadratic functions.

Unveiling the Shared Roots The X-Intercepts

When we talk about x-intercepts, we're referring to the points where the parabola intersects the x-axis. These are the points where the function's value, or y, equals zero. For a parabola expressed in factored form, like our examples, the x-intercepts are readily identifiable. The factored form of a quadratic equation, such as f(x) = a(x - r)(x - s), directly reveals the roots or x-intercepts of the parabola, which are 'r' and 's'. Let's examine f(x) = 2(x + 8)(x - 1). To find its x-intercepts, we set f(x) to zero: 2(x + 8)(x - 1) = 0. This equation holds true when either (x + 8) = 0 or (x - 1) = 0. Solving these equations gives us x = -8 and x = 1. These are the x-intercepts of the parabola f(x). Now, let's consider g(x) = -3/4(x + 8)(x - 1). We repeat the process, setting g(x) to zero: -3/4(x + 8)(x - 1) = 0. Again, this equation is satisfied when either (x + 8) = 0 or (x - 1) = 0, leading to the same solutions: x = -8 and x = 1. Crucially, both parabolas share the same x-intercepts: x = -8 and x = 1. This shared characteristic arises directly from the identical factors (x + 8) and (x - 1) present in both equations. The coefficients 2 and -3/4 only affect the parabola's vertical stretch or compression and its direction (whether it opens upwards or downwards) but do not alter the x-intercepts. In essence, the x-intercepts are determined solely by the roots of the quadratic equation, which are the values of x that make the function equal to zero. This fundamental concept is a cornerstone of understanding quadratic functions and their graphical representations. The x-intercepts provide valuable information about the parabola's position and its relationship to the x-axis. Knowing the x-intercepts is often the first step in sketching a parabola or solving related problems.

Contrasting the Curves Vertex and Direction

While the x-intercepts remain consistent, other features of these parabolas diverge. Let's consider the vertex. The vertex represents the minimum or maximum point of the parabola, depending on whether it opens upwards or downwards. The x-coordinate of the vertex lies exactly midway between the x-intercepts. For both f(x) and g(x), the x-intercepts are -8 and 1. The midpoint, and therefore the x-coordinate of the vertex, is (-8 + 1) / 2 = -3.5. However, the y-coordinate of the vertex depends on the leading coefficient in the equation. To find the y-coordinate, we substitute x = -3.5 into each equation. For f(x) = 2(x + 8)(x - 1), f(-3.5) = 2(-3.5 + 8)(-3.5 - 1) = 2(4.5)(-4.5) = -40.5. Thus, the vertex of f(x) is (-3.5, -40.5). For g(x) = -3/4(x + 8)(x - 1), g(-3.5) = -3/4(-3.5 + 8)(-3.5 - 1) = -3/4(4.5)(-4.5) = 15.1875. The vertex of g(x) is (-3.5, 15.1875). As we can see, despite sharing the same x-coordinate, the vertices of the two parabolas have different y-coordinates. This difference arises from the different leading coefficients in the equations, which cause vertical stretching or compression and reflection. Next, let's examine the direction in which the parabolas open. A parabola opens upwards if its leading coefficient (the coefficient of the x² term) is positive and downwards if it is negative. In f(x) = 2(x + 8)(x - 1), the leading coefficient is 2, which is positive. Therefore, f(x) opens upwards. In g(x) = -3/4(x + 8)(x - 1), the leading coefficient is -3/4, which is negative. Consequently, g(x) opens downwards. This difference in direction is a direct consequence of the sign of the leading coefficient. A positive leading coefficient results in a parabola that has a minimum value (the vertex), while a negative leading coefficient results in a parabola that has a maximum value (the vertex). These contrasting characteristics highlight the importance of the leading coefficient in determining the overall shape and orientation of a parabola.

Delving into the Y-Intercepts

Another distinguishing feature of these parabolas is their y-intercepts. The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. To find the y-intercept, we substitute x = 0 into each equation. For f(x) = 2(x + 8)(x - 1), f(0) = 2(0 + 8)(0 - 1) = 2(8)(-1) = -16. The y-intercept of f(x) is (0, -16). For g(x) = -3/4(x + 8)(x - 1), g(0) = -3/4(0 + 8)(0 - 1) = -3/4(8)(-1) = 6. The y-intercept of g(x) is (0, 6). Clearly, the y-intercepts of the two parabolas are different. This difference arises because the constant term in the expanded form of the quadratic equation determines the y-intercept. When we expand the equations, we get: f(x) = 2(x² + 7x - 8) = 2x² + 14x - 16 g(x) = -3/4(x² + 7x - 8) = -3/4x² - 21/4x + 6 The constant terms, -16 and 6, correspond to the y-coordinates of the y-intercepts. The y-intercept provides valuable information about the parabola's vertical position relative to the y-axis. It represents the point where the parabola begins its curve, either upwards or downwards. Understanding the y-intercept, along with the x-intercepts and vertex, provides a comprehensive picture of the parabola's shape and location on the coordinate plane. In summary, while the parabolas f(x) and g(x) share the same x-intercepts, their y-intercepts differ due to the different leading coefficients and the resulting vertical scaling and reflection. This further underscores the role of the leading coefficient in shaping the parabola's overall characteristics.

Conclusion: Shared Intercepts, Divergent Paths

In conclusion, the parabolas f(x) = 2(x + 8)(x - 1) and g(x) = -3/4(x + 8)(x - 1) share the same x-intercepts, a consequence of their common factors (x + 8) and (x - 1). However, they differ in their vertices, directions of opening, and y-intercepts. f(x) opens upwards with a vertex at (-3.5, -40.5) and a y-intercept of (0, -16), while g(x) opens downwards with a vertex at (-3.5, 15.1875) and a y-intercept of (0, 6). These differences stem from the distinct leading coefficients in their equations, which affect vertical stretch, compression, reflection, and ultimately, the overall shape and position of the parabolas. Understanding these distinctions is crucial for comprehending the behavior of quadratic functions and their graphical representations. By analyzing the equations and identifying key features such as intercepts, vertices, and direction of opening, we can gain a deeper appreciation for the diverse characteristics of parabolas.