Quadratic Functions: Simplifying Polynomial Expressions

When delving into the fascinating world of algebra, you'll often encounter different types of functions. Today, we're going to explore how combining two linear functions, $a(x) = 2x - 4$ and $b(x) = x + 2$, can sometimes result in a quadratic function. A quadratic function is a polynomial function of degree two, meaning its highest power of the variable is 2. These functions typically have a characteristic U-shaped or inverted U-shaped graph called a parabola. Understanding how to manipulate and combine these functions is a fundamental skill in mathematics, opening doors to solving more complex problems and understanding advanced concepts. We'll break down each potential expression to see which one yields this desirable quadratic form. Get ready to simplify, multiply, and divide your way to understanding!

Understanding the Basics: Linear vs. Quadratic Functions

Before we dive into combining our functions, let's clarify what makes a function linear or quadratic. A linear function is a function whose graph is a straight line. It can be represented in the form $f(x) = mx + c$, where $m$ is the slope and $c$ is the y-intercept. Both $a(x) = 2x - 4$ and $b(x) = x + 2$ are classic examples of linear functions. Their graphs are straight lines with different slopes and intercepts. On the other hand, a quadratic function is a function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and importantly, $a$ cannot be zero. If $a$ were zero, the $x^2$ term would disappear, and the function would revert to being linear. The presence of the $x^2$ term is what gives quadratic functions their unique parabolic shape and their second-degree status. Recognizing these differences is crucial when we start performing operations like addition, subtraction, multiplication, and division on our given functions, $a(x)$ and $b(x)$. Our goal is to find an operation that, when applied to these two linear functions, will introduce that $x^2$ term, thus transforming the result into a quadratic function. It's like a mathematical puzzle where we're looking for the right key to unlock a specific type of function.

Analyzing the Operations: Step-by-Step Exploration

Let's systematically examine each of the given expressions to determine which one results in a quadratic function. We have our two linear functions: $a(x) = 2x - 4$ and $b(x) = x + 2$. We need to see what happens when we perform the operations $(a imes b)(x)$, (rac{a}{b})(x), $(a - b)(x)$, and $(a + b)(x)$.

1. The Product: $(a imes b)(x)$

This expression represents the multiplication of the two functions $a(x)$ and $b(x)$. To find $(a imes b)(x)$, we simply multiply the expressions for $a(x)$ and $b(x)$ together:

$(a imes b)(x) = a(x) imes b(x)$ $(a imes b)(x) = (2x - 4)(x + 2)$

To expand this, we can use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last):

• First: $(2x)(x) = 2x^2$
• Outer: $(2x)(2) = 4x$
• Inner: $(-4)(x) = -4x$
• Last: $(-4)(2) = -8$

Now, we combine these terms:

$(a imes b)(x) = 2x^2 + 4x - 4x - 8$

Simplifying by combining the like terms ($4x$ and $-4x$ cancel each other out):

$(a imes b)(x) = 2x^2 - 8$

Looking at this resulting expression, $2x^2 - 8$, we can see that the highest power of $x$ is 2. This fits the definition of a quadratic function. Therefore, the multiplication of $a(x)$ and $b(x)$ produces a quadratic function. This is a key discovery in our exploration!

2. The Quotient: $\left(\frac{a}{b}\right)(x)$

This expression represents the division of function $a(x)$ by function $b(x)$.

$\left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)}$ $\left(\frac{a}{b}\right)(x) = \frac{2x - 4}{x + 2}$

This is a rational function. Unless there's significant simplification possible that eliminates the variable from the denominator (which is not the case here, as $x+2$ is not a factor of $2x-4$ in a way that cancels the $x$), this expression will not be a polynomial, let alone a quadratic function. The degree of the numerator is 1, and the degree of the denominator is 1. Division of polynomials does not generally result in a polynomial of a specific degree unless under very specific circumstances. In this case, it's a rational expression, not a quadratic function.

3. The Difference: $(a - b)(x)$

This expression represents the subtraction of function $b(x)$ from function $a(x)$.

$(a - b)(x) = a(x) - b(x)$ $(a - b)(x) = (2x - 4) - (x + 2)$

Remember to distribute the negative sign to both terms in $b(x)$:

$(a - b)(x) = 2x - 4 - x - 2$

Now, combine like terms:

• Combine $x$ terms: $2x - x = x$
• Combine constant terms: $-4 - 2 = -6$

So, $(a - b)(x) = x - 6$

This resulting expression, $x - 6$, is a linear function because the highest power of $x$ is 1. It does not produce a quadratic function.

4. The Sum: $(a + b)(x)$

This expression represents the addition of function $a(x)$ and function $b(x)$.

$(a + b)(x) = a(x) + b(x)$ $(a + b)(x) = (2x - 4) + (x + 2)$

Remove the parentheses:

$(a + b)(x) = 2x - 4 + x + 2$

Combine like terms:

• Combine $x$ terms: $2x + x = 3x$
• Combine constant terms: $-4 + 2 = -2$

So, $(a + b)(x) = 3x - 2$

Similar to the difference, this resulting expression, $3x - 2$, is also a linear function because the highest power of $x$ is 1. It does not produce a quadratic function.

The Verdict: Which Expression Yields a Quadratic Function?

After carefully analyzing each operation, we found that:

• $(a imes b)(x) = 2x^2 - 8$ (This is a quadratic function)
• $\left(\frac{a}{b}\right)(x) = \frac{2x - 4}{x + 2}$ (This is a rational function)
• $(a - b)(x) = x - 6$ (This is a linear function)
• $(a + b)(x) = 3x - 2$ (This is a linear function)

Therefore, the expression that produces a quadratic function is $(a imes b)(x)$. This is because multiplying two linear expressions, especially when they don't share common factors that would cancel out terms, often results in a second-degree polynomial. The multiplication introduces the $x^2$ term, which is the hallmark of a quadratic function. It's a great example of how different algebraic operations can change the nature and degree of a function.

Why Multiplication is Key for Quadratics Here

The reason why the product $(a imes b)(x)$ yields a quadratic function, while the sum, difference, and quotient do not, lies in the fundamental properties of polynomial degrees. When you add or subtract polynomials, the degree of the resulting polynomial is at most the highest degree of the original polynomials. For example, adding or subtracting two linear functions (degree 1) will always result in another linear function (degree 1) or a constant (degree 0), unless the $x$ terms cancel out perfectly, leaving only a constant. In our case, $(2x - 4) + (x + 2) = 3x - 2$ and $(2x - 4) - (x + 2) = x - 6$. Both have degree 1. The quotient of two polynomials results in a rational function, which is not a polynomial unless the denominator divides the numerator evenly and the result is a polynomial. Division by $x+2$ doesn't simplify rac{2x-4}{x+2} into a polynomial. However, when you multiply polynomials, the degree of the product is the sum of the degrees of the original polynomials. Since both $a(x)$ and $b(x)$ are degree 1 (linear functions), their product will have a degree of $1 + 1 = 2$. This is precisely the definition of a quadratic function. The process of multiplication creates the $x^2$ term, which is essential for a function to be classified as quadratic. This mathematical principle is a cornerstone of algebra and is crucial for understanding how polynomial degrees behave under different operations. It highlights the power of multiplication in elevating the degree of a polynomial expression.

Conclusion: Mastering Function Combinations

In conclusion, when given the linear functions $a(x) = 2x - 4$ and $b(x) = x + 2$, the expression that produces a quadratic function is $(a imes b)(x)$. This is because the multiplication of two degree-1 polynomials inherently results in a degree-2 polynomial, which is the definition of a quadratic function. The other operations, addition, subtraction, and division, in this specific case, result in linear or rational functions, respectively. Mastering these fundamental operations is essential for anyone studying mathematics, as it lays the groundwork for understanding more complex functions and problem-solving techniques. It’s all about understanding how the degree of a polynomial changes (or doesn’t change) with each arithmetic operation.

For further exploration into the fascinating world of quadratic functions and their properties, you can visit resources like Khan Academy. They offer comprehensive guides and practice problems that can deepen your understanding of these important mathematical concepts.