Finding X-Intercepts: A Guide To Quadratic Functions

Hey everyone! Let's dive into a classic math problem that often pops up: finding the x-intercepts of a quadratic function. Specifically, we'll be tackling the question, "Which point is an x-intercept of the quadratic function f(x) = (x + 6)(x - 3)?" Don't worry, it's not as scary as it might sound! We'll break it down step by step. Understanding x-intercepts is super important when you're dealing with quadratic functions, as they tell us where the graph crosses the x-axis. This is crucial for graphing the function and understanding its behavior. This concept is also fundamental to solving quadratic equations, so nailing this down is a win-win! We'll explain what an x-intercept is, how to find it, and then apply that knowledge to the given function. So, let's get started and make sure you understand this concept like a pro. Let's explore why finding the x-intercepts matters and how they relate to the real-world applications of quadratic functions. This knowledge can be helpful in many scenarios, from simple calculations to complex problem-solving. Ready to become x-intercept masters? Let's get started, guys!

Understanding X-Intercepts: What's the Deal?

Okay, so first things first: what exactly is an x-intercept? Think of it this way: the x-intercept is simply the point where a graph crosses the x-axis. At this point, the y-coordinate is always zero. Always! This is because any point on the x-axis has a y-value of zero. Understanding this is super important because it's the foundation for everything else we're going to talk about. An x-intercept is also called a root or a zero of the function. These terms are used interchangeably, so get used to seeing them. When you're looking for an x-intercept, you're essentially trying to find the value(s) of x that make the function equal to zero. So, the x-intercept is the x-value when f(x) = 0. Got it? Let's make it super clear. Consider a simple linear equation, y = 2x + 4. To find the x-intercept, we set y = 0 and solve for x. This gives us 0 = 2x + 4, which simplifies to x = -2. Therefore, the x-intercept is the point (-2, 0). Every time you see the term "x-intercept", remember that it means the point where the graph of the function meets the x-axis, and the y-value at that point is always 0. The concept of x-intercepts extends beyond just lines and can be applied to any function, especially quadratic functions, which are the focus of this problem. Being able to identify x-intercepts is crucial for understanding the behavior of the function and its visual representation on a graph. This understanding is also useful in practical applications. For example, x-intercepts can help find the break-even point in business or determine the time at which an object hits the ground in physics. So, keep in mind the core concept: x-intercepts are where y=0, and you're golden!

Solving for X-Intercepts in the Given Quadratic Function

Now that we've got the basics down, let's tackle the question. We're given the quadratic function f(x) = (x + 6)(x - 3). Our goal is to find the x-intercept(s). Remember, finding the x-intercept means finding the value(s) of x that make f(x) = 0. In other words, we need to solve the equation (x + 6)(x - 3) = 0. This is where things get pretty straightforward, thanks to the zero-product property. The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. So, to solve (x + 6)(x - 3) = 0, we set each factor equal to zero and solve for x. First, we have x + 6 = 0. Solving for x, we get x = -6. This means that when x = -6, f(x) = 0. Second, we have x - 3 = 0. Solving for x, we get x = 3. This means that when x = 3, f(x) = 0. Therefore, the x-intercepts of the quadratic function are x = -6 and x = 3. Thus, our graph intersects the x-axis at two points. These points can be written as coordinates: (-6, 0) and (3, 0). The question asks us to find a single point which is an x-intercept. Let's revisit the options in the question to find the correct one. Using the zero-product property makes it very easy to solve this kind of problem. It's all about breaking the quadratic function down into its factors and solving for x. With a little practice, you'll be able to solve similar problems with ease, no sweat! This step helps you understand how to apply the zero-product property to find the x-intercepts of this quadratic equation.

Matching the X-Intercepts with the Given Options

Now, let's go back to the options provided in the question. We've determined that the x-intercepts are the points where x = -6 and x = 3. Remember, the y-coordinate at any x-intercept is always 0. That means that the correct x-intercepts, written as coordinate points, are (-6, 0) and (3, 0). Now, let's look at the given choices:

• A. (0, 6): This option is incorrect. The x-intercepts do not have an x-coordinate of 0. The y-coordinate being 6 does not fit with what we know.
• B. (0, -6): This option is also incorrect. Like option A, this point's x-coordinate isn't what we're looking for, as it does not align with what we found.
• C. (6, 0): This option is incorrect because, although the y-coordinate is correct, the x-coordinate we found for an intercept doesn't align with the option's.
• D. (-6, 0): This is the correct option. We determined that one of the x-intercepts is at x = -6, and at the x-intercept, the y-coordinate is always 0. Therefore, the point (-6, 0) is an x-intercept of the given quadratic function.

So, the correct answer is D! See? Not so bad, right? By applying our knowledge of what an x-intercept is and how to find it using the zero-product property, we were able to quickly identify the correct answer. This method of solving the problems ensures we can arrive at the right answers and understand the steps along the way.

Conclusion: Mastering X-Intercepts

Awesome work, everyone! We've successfully identified the x-intercept of the quadratic function f(x) = (x + 6)(x - 3). We've covered the basics, from understanding what an x-intercept is to applying the zero-product property to find the x-intercepts of a quadratic function. Understanding x-intercepts is fundamental in algebra. Knowing how to find x-intercepts will give you a better grasp of how quadratic functions work, how they're graphed, and how you can solve equations using them. The concept of x-intercepts also paves the way for understanding other important concepts, like the vertex of a parabola and how to solve quadratic equations by factoring. The zero-product property is a powerful tool that comes in handy when dealing with quadratic equations. Keep practicing, and you'll become a pro in no time. Remember, the x-intercept is the point where the graph meets the x-axis, making the y-coordinate equal to zero. And, with the zero-product property, you can quickly find these points. Thanks for sticking around, and happy math-ing, guys! Keep up the great work!