Identifying Quadratic Functions From Combined Expressions

Hey guys! Let's dive into a fascinating mathematical puzzle that involves spotting quadratic functions hidden within algebraic expressions. This is a common type of question you might encounter in algebra, and understanding the nuances can significantly boost your problem-solving skills. We're going to dissect the problem piece by piece, so buckle up and let's get started!

The Core Question: Spotting the Quadratic

So, the main question we're tackling today is: "If $a(x) = 2x - 4$ and $b(x) = x + 2$, which of the following expressions produces a quadratic function?"

• A. $(ab)(x)$
• B. $(\frac{a}{b})(x)$
• C. $(a-b)(x)$
• D. $(a+b)(x)$

Sounds intriguing, right? At first glance, it might seem a bit overwhelming, but trust me, with a systematic approach, it's totally manageable. The key here is to understand what a quadratic function is and how different operations on functions affect their degree. So let's begin with understanding a quadratic function and its key characteristics.

Unpacking Quadratic Functions: The Basics

Before we jump into solving the problem, let's make sure we're all on the same page about what a quadratic function actually is. In essence, a quadratic function is a polynomial function of degree two. What does that mean? It means the highest power of the variable (in this case, x) is 2. The general form of a quadratic function is:

$f(x) = ax^2 + bx + c$

Where a, b, and c are constants, and a is not equal to zero (otherwise, it would just be a linear function). The $ax^2$ term is what makes it quadratic. Key Characteristics of Quadratic Functions include:

• The graph of a quadratic function is a parabola, a U-shaped curve. This parabolic shape is a visual indicator of a quadratic function.
• The degree of the polynomial is 2, meaning the highest power of the variable x is 2. This is the most crucial aspect to identify a quadratic function.
• Quadratic functions often appear in various real-world applications, such as projectile motion, optimization problems, and curve fitting. Recognizing them is vital in practical scenarios.

Now that we've refreshed our understanding of quadratic functions, we can approach the problem more strategically. Remember, we're looking for an expression that, after simplification, results in a function with an $x^2$ term. Let’s keep this in mind as we evaluate each option.

Option A: $(ab)(x)$ - The Product of Two Functions

Let's kick things off by examining option A: $(ab)(x)$. This notation represents the product of the two functions, $a(x)$ and $b(x)$. In simpler terms, we need to multiply the expressions for $a(x)$ and $b(x)$ together. Given that $a(x) = 2x - 4$ and $b(x) = x + 2$, we can write:

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

To find the resulting function, we need to expand this product. We can use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last) to multiply the two binomials:

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

Now, let's combine these terms:

$(ab)(x) = 2x^2 + 4x - 4x - 8$

Notice that the $4x$ and $-4x$ terms cancel each other out, which simplifies the expression to:

$(ab)(x) = 2x^2 - 8$

Aha! We have an $x^2$ term, which means this is indeed a quadratic function. So, option A looks promising. But just to be thorough, let's examine the other options to make sure we haven't missed anything. Understanding why other options are not quadratic is as important as identifying the correct one.

Option B: $(\frac{a}{b})(x)$ - The Quotient of Two Functions

Moving on to option B, we have $(\frac{a}{b})(x)$. This represents the quotient of the two functions, $a(x)$ divided by $b(x)$. So, we need to divide the expression for $a(x)$ by the expression for $b(x)$. Given $a(x) = 2x - 4$ and $b(x) = x + 2$, we have:

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

Now, let's see if we can simplify this expression. Notice that we can factor out a 2 from the numerator:

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

At this point, we can see that there are no further simplifications possible. The expression remains a rational function, which is a ratio of two polynomials. Importantly, there is no $x^2$ term in the simplified expression. Rational functions do not generally represent quadratic functions unless they can be simplified to a quadratic form, which is not the case here. Therefore, Option B is not a quadratic function. Recognizing the structure of different types of functions is key here. So, Option B is not the correct answer.

Option C: $(a-b)(x)$ - The Difference of Two Functions

Let's consider option C, $(a-b)(x)$. This represents the difference between the two functions, $a(x)$ and $b(x)$. To find this, we subtract the expression for $b(x)$ from the expression for $a(x)$. Given $a(x) = 2x - 4$ and $b(x) = x + 2$, we can write:

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

Remember to distribute the negative sign to both terms inside the parentheses:

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

Now, combine like terms:

$(a-b)(x) = (2x - x) + (-4 - 2) = x - 6$

The resulting function is $x - 6$, which is a linear function (a polynomial of degree 1). There is no $x^2$ term, so this is not a quadratic function. Linear functions have a straight-line graph, unlike the parabolic graph of a quadratic function. Thus, Option C can be eliminated.

Option D: $(a+b)(x)$ - The Sum of Two Functions

Finally, let's examine option D, $(a+b)(x)$. This represents the sum of the two functions, $a(x)$ and $b(x)$. To find this, we add the expressions for $a(x)$ and $b(x)$ together. Given $a(x) = 2x - 4$ and $b(x) = x + 2$, we can write:

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

Now, combine like terms:

$(a+b)(x) = (2x + x) + (-4 + 2) = 3x - 2$

The resulting function is $3x - 2$, which, like option C, is a linear function. There's no $x^2$ term present. Therefore, this is not a quadratic function either. Adding two linear functions will always result in another linear function (unless the x terms cancel out, resulting in a constant function).

The Verdict: Option A is the Winner!

After carefully analyzing all the options, we've come to a conclusion. Let's recap what we found:

• Option A, $(ab)(x)$, resulted in the quadratic function $2x^2 - 8$. This is because multiplying two linear expressions can yield a quadratic expression.
• Option B, $(\frac{a}{b})(x)$, resulted in a rational function, not a quadratic.
• Option C, $(a-b)(x)$, resulted in a linear function, $x - 6$.
• Option D, $(a+b)(x)$, resulted in a linear function, $3x - 2$.

Therefore, the expression that produces a quadratic function is indeed option A. Great job, guys! You've successfully navigated through this problem by understanding the properties of quadratic functions and how different operations affect the degree of the resulting expression.

Final Thoughts: Mastering Function Operations

This question highlights the importance of understanding function operations – addition, subtraction, multiplication, and division – and how they transform the nature of functions. Being able to quickly identify a quadratic function and distinguish it from linear and rational functions is a valuable skill in algebra and calculus.

Key takeaways from this exploration include:

• A quadratic function has the general form $f(x) = ax^2 + bx + c$, where a is not zero.
• Multiplying two linear functions can result in a quadratic function.
• Adding or subtracting linear functions results in a linear function.
• Dividing linear functions results in a rational function, not a quadratic function.

Keep practicing these types of problems, and you'll become a pro at spotting quadratic functions in no time. Remember, math is like a puzzle – each piece fits together to create a beautiful solution. Keep exploring and keep learning!