Identifying Quadratic Functions From Linear Function Operations A(x) = 2x - 4 And B(x) = X + 2
When exploring quadratic functions, it's essential to understand how different operations on linear functions can lead to the formation of quadratic expressions. In this article, we will delve into the scenario where we have two linear functions, a(x) = 2x - 4 and b(x) = x + 2, and examine which operations between these functions result in a quadratic function. A quadratic function is defined as a polynomial function of degree two, generally represented in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠0. The key characteristic of a quadratic function is the presence of the x² term, which determines the parabolic shape of its graph. Therefore, to identify which expression yields a quadratic function, we need to perform the indicated operations and observe whether an x² term appears in the resulting expression.
Analyzing the Options
Let's consider the given options and determine which one produces a quadratic function:
A. (ab)(x)
To find (ab)(x), we need to multiply the two functions a(x) and b(x). This operation involves multiplying the expressions representing a(x) and b(x), which is a straightforward application of the distributive property. This multiplication will reveal if a quadratic term, specifically an x² term, is generated, thus indicating whether the resulting function is quadratic. Understanding how polynomial multiplication works is crucial here, as it directly affects the degree of the resulting polynomial. If the highest degree term is x², then we can definitively classify the resulting function as quadratic.
So, let's multiply a(x) and b(x):
(ab)(x) = a(x) * b(x) = (2x - 4)(x + 2)
Now, we apply the distributive property (also known as the FOIL method) to expand this product:
(2x - 4)(x + 2) = 2x(x) + 2x(2) - 4(x) - 4(2) = 2x² + 4x - 4x - 8
Notice that the 4x and -4x terms cancel each other out, simplifying the expression:
= 2x² - 8
The resulting expression is 2x² - 8, which clearly includes an x² term. This confirms that the product of a(x) and b(x) is a quadratic function because it fits the form ax² + bx + c, where a = 2, b = 0, and c = -8. The presence of the 2x² term is the key indicator here, marking this result as a quadratic expression. Therefore, option A is a strong candidate for producing a quadratic function.
B. (a/b)(x)
For the expression (a/b)(x), we need to divide a(x) by b(x). This operation involves forming a rational function, which is a ratio of two polynomials. The crucial aspect here is to determine whether the resulting function will be a polynomial function, and if so, what its degree will be. Dividing polynomials can sometimes result in a simplified polynomial expression, but it often leads to a rational function that is not a polynomial, especially if the denominator does not completely divide the numerator. Understanding the rules of polynomial division and simplification is key to analyzing this option.
Let's perform the division:
(a/b)(x) = a(x) / b(x) = (2x - 4) / (x + 2)
To simplify this expression, we can try to factor the numerator and see if there are any common factors with the denominator:
2x - 4 = 2(x - 2)
So, the expression becomes:
(a/b)(x) = 2(x - 2) / (x + 2)
In this case, there are no common factors between the numerator and the denominator that can be cancelled out. Therefore, the expression remains a rational function. This rational function is not a quadratic function because it's not a polynomial of degree two. Instead, it's a ratio of two linear functions, which generally results in a hyperbolic-shaped graph, not a parabola. Thus, option B does not produce a quadratic function.
C. (a-b)(x)
To find (a - b)(x), we need to subtract the function b(x) from a(x). This operation involves subtracting one linear expression from another, which is a fundamental algebraic operation. The result of subtracting linear functions will always be another linear function, unless the leading terms cancel out completely, resulting in a constant function. Understanding how subtraction affects the terms of polynomials is crucial for determining the nature of the resulting function. Since quadratic functions require an x² term, subtracting linear functions is unlikely to produce one.
Let's subtract b(x) from a(x):
(a - b)(x) = a(x) - b(x) = (2x - 4) - (x + 2)
Now, we distribute the negative sign and combine like terms:
= 2x - 4 - x - 2 = (2x - x) + (-4 - 2) = x - 6
The resulting expression is x - 6, which is a linear function. This confirms that subtracting a linear function from another linear function results in another linear function. There is no x² term, so (a - b)(x) is not a quadratic function. Therefore, option C is not the correct answer.
D. (a+b)(x)
For the expression (a + b)(x), we need to add the two functions a(x) and b(x). This operation involves combining two linear expressions, which is a basic algebraic operation. Similar to subtraction, adding linear functions will typically result in another linear function, unless the terms combine in such a way that they simplify to a constant. The absence of any multiplication or higher-order operations means that an x² term, which is necessary for a quadratic function, is unlikely to appear. Understanding the rules of polynomial addition is key to predicting the outcome of this operation.
Let's add a(x) and b(x):
(a + b)(x) = a(x) + b(x) = (2x - 4) + (x + 2)
Now, we combine like terms:
= (2x + x) + (-4 + 2) = 3x - 2
The resulting expression is 3x - 2, which is a linear function. This clearly shows that adding two linear functions results in another linear function. There is no x² term present, indicating that (a + b)(x) is not a quadratic function. Therefore, option D is not the correct answer.
Conclusion
After analyzing all the options, we can definitively conclude that only option A, (ab)(x), produces a quadratic function. This is because multiplying the two linear functions a(x) and b(x) results in an expression with an x² term, which is the defining characteristic of a quadratic function. The other options, involving division, subtraction, and addition, do not yield a quadratic function as they do not produce the necessary x² term.
In summary, when dealing with operations on functions, it's crucial to understand how these operations affect the degree of the resulting function. Multiplication can increase the degree, while addition and subtraction typically maintain the degree. Division can lead to rational functions rather than polynomial functions. This analysis helps in identifying the specific operations that lead to the formation of quadratic functions from linear functions.
