(x-3)(x-4).jpeg "(x+1)(x-3)(x-4)")
Exploring the Cubic Function: (x+1)(x-3)(x-4)
This article will delve into the properties and characteristics of the cubic function represented by the expression (x+1)(x-3)(x-4). Understanding this function provides insights into its behavior, its graph, and its roots.
Factorized Form and Roots
The given expression is already in its factored form. This makes it easy to identify the roots of the function. A root is a value of x that makes the function equal to zero.
- x+1 = 0 implies x = -1
- x-3 = 0 implies x = 3
- x-4 = 0 implies x = 4
Therefore, the function has three roots: -1, 3, and 4.
Expanding the Expression
To understand the function's behavior further, we can expand the expression:
(x+1)(x-3)(x-4) = (x^2 - 2x - 3)(x-4)
= x^3 - 6x^2 + 5x + 12
The expanded form x³ - 6x² + 5x + 12 provides information about the function's degree, leading coefficient, and constant term.
- Degree: The highest power of x is 3, indicating a cubic function.
- Leading Coefficient: The coefficient of the x³ term is 1, which is positive. This means the function will have a positive leading behavior.
- Constant Term: The constant term is 12, which represents the y-intercept of the function's graph.
Graphing the Function
The graph of this cubic function will exhibit the following characteristics:
- Roots: The graph will intersect the x-axis at the points (-1, 0), (3, 0), and (4, 0).
- Y-Intercept: The graph will intersect the y-axis at the point (0, 12).
- Leading Behavior: Since the leading coefficient is positive, the graph will rise towards positive infinity as x approaches positive infinity and fall towards negative infinity as x approaches negative infinity.
The graph will have a local maximum between the roots -1 and 3, and a local minimum between the roots 3 and 4.
Conclusion
The cubic function represented by (x+1)(x-3)(x-4) has three roots, a positive leading coefficient, and a constant term of 12. Its graph will exhibit a specific shape with local maxima and minima, and will intersect the x-axis at the roots and the y-axis at the constant term. By analyzing its factored form, expanded form, and graph, we gain valuable insights into the behavior of this cubic function.