Finding X-Intercepts: F(x) = (x-8)(x+9) Explained

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Finding X-Intercepts of Quadratic Functions: A Step-by-Step Guide

Hey guys! Let's dive into a common question in mathematics: finding the x-intercepts of a quadratic function. This is a fundamental concept in algebra, and understanding it will help you tackle various problems. We'll break down the process using a specific example, so you can see exactly how it's done. So, let's get started and make sure you're crystal clear on how to find those x-intercepts!

Understanding X-Intercepts

First off, what exactly is an x-intercept? Simply put, an x-intercept is the point where a graph crosses the x-axis. At this point, the y-value (or f(x) value, in function notation) is always zero. Think of it like this: you're on the x-axis, so you haven't moved up or down at all. When we talk about finding x-intercepts, we're essentially asking: for what values of x does the function equal zero?

In the context of a quadratic function, which is a function of the form f(x) = ax² + bx + c, the x-intercepts are also known as the roots or zeros of the function. A quadratic function can have two, one, or no real x-intercepts, depending on how the parabola intersects the x-axis. Finding these intercepts is crucial in understanding the behavior and characteristics of the quadratic function. These points tell us where the parabola starts, stops, or changes direction in relation to the x-axis. The x-intercepts are invaluable for sketching the graph, solving equations, and understanding real-world applications modeled by quadratic functions.

To find the x-intercepts, we set f(x) equal to zero and solve for x. This is because, as mentioned earlier, the y-value is zero at the x-intercept. This process often involves factoring the quadratic equation, using the quadratic formula, or completing the square. Each of these methods allows us to transform the equation into a form where we can easily identify the values of x that make the function equal to zero. Once we find these values, we express them as coordinate points (x, 0), which represent the exact locations where the parabola intersects the x-axis. So, understanding x-intercepts is a cornerstone for mastering quadratic functions and their applications.

The Problem: f(x) = (x-8)(x+9)

Now, let's tackle a specific problem. We're given the quadratic function f(x) = (x-8)(x+9), and our mission is to find its x-intercepts. We're also given multiple-choice options: A. (0,8), B. (0,-8), C. (9,0), and D. (-9,0). Only one of these is the correct answer. This means we need to find the x values that make the function equal to zero.

Why is this in factored form? Well, it's actually a big help! When a quadratic function is in factored form, like ours, finding the x-intercepts becomes much simpler. The factored form directly shows us the values of x that will make each factor equal to zero, and therefore make the entire function equal to zero. This is because of the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is a cornerstone for solving equations in factored form. For example, if we have (a)(b) = 0, then either a = 0 or b = 0 (or both). In our case, the factors are (x-8) and (x+9), so we need to determine the values of x that make each of these factors equal to zero.

The factored form of a quadratic equation not only simplifies the process of finding x-intercepts but also provides valuable insights into the graph of the parabola. The factors directly correspond to the x-intercepts, which are the points where the parabola crosses the x-axis. Knowing these points allows us to quickly sketch the parabola or understand its behavior without having to plot numerous points. Additionally, the factored form can help us determine the axis of symmetry, which is the vertical line that passes through the vertex of the parabola. The x-coordinate of the vertex is the average of the x-intercepts, providing a straightforward way to find the middle point of the parabola. This form is particularly useful in various applications, such as solving word problems involving projectile motion or optimization, where identifying the roots of the equation is essential. So, understanding and working with factored form is a powerful tool in analyzing quadratic functions.

Solving for the X-Intercepts

To find the x-intercepts, we set f(x) = 0. So, we have the equation 0 = (x-8)(x+9). Now, we apply the zero-product property. This means that either (x-8) = 0 or (x+9) = 0. We've broken down the problem into two simpler equations.

Let's solve each one separately.

First, take x - 8 = 0. Adding 8 to both sides of the equation, we get x = 8. This means one of our x-intercepts occurs when x is 8. This is a crucial step because it identifies one of the points where the parabola intersects the x-axis. By isolating x, we’ve determined the x-coordinate of this intercept, which is a key component in understanding the graph’s behavior. This value represents a root of the quadratic equation and helps in visualizing where the parabola crosses the x-axis. Understanding how to solve such equations is vital for analyzing quadratic functions and their real-world applications.

Next, we solve x + 9 = 0. Subtracting 9 from both sides, we find x = -9. This is our second x-intercept. Just like before, this x-value indicates another point where the parabola crosses the x-axis. The negative value tells us that this intercept is located to the left of the y-axis. The process of isolating x here reinforces the fundamental algebraic principle of maintaining balance in an equation. This x-intercept is equally important as the first one in sketching the graph of the parabola and understanding its position in the coordinate plane. Both x-intercepts provide critical information about the quadratic function's roots and its graphical representation.

So, we have two x-values: x = 8 and x = -9. Remember, x-intercepts are points on the graph, so we express them as coordinate pairs. Since the y-value is always 0 at the x-intercept, our points are (8, 0) and (-9, 0). These coordinate pairs represent the exact locations where the parabola intersects the x-axis. Each point is a solution to the equation f(x) = 0 and is a critical component in understanding the quadratic function’s behavior. The x-coordinates 8 and -9 are the roots or zeros of the function, and they provide valuable information for sketching the parabola and solving related problems. The clarity in expressing these solutions as coordinate pairs ensures we understand the graphical representation of the x-intercepts and their significance in the context of the quadratic function.

Choosing the Correct Answer

Now, let's look back at our multiple-choice options:

A. (0,8)

B. (0,-8)

C. (9,0)

D. (-9,0)

We found the x-intercepts to be (8, 0) and (-9, 0). Comparing these to the options, we see that option D, (-9,0), is one of our solutions. Option C, (9,0), is close, but it's not quite right. Remember, the sign is important! The other options, A and B, have the x and y coordinates switched and are therefore incorrect.

Why are options A and B incorrect? It's because they represent y-intercepts, not x-intercepts. A y-intercept is the point where the graph crosses the y-axis, meaning the x-value is zero. In these options, x is 0, and the y-values are 8 and -8, respectively. While y-intercepts are important features of a graph, they are different from x-intercepts, which we were specifically asked to find. Understanding the distinction between x and y-intercepts is crucial for correctly interpreting the graph and its behavior. Confusing these intercepts can lead to incorrect solutions and a misunderstanding of the function’s properties. Therefore, it’s essential to always verify which intercept the question is asking for and to ensure the correct coordinate pair is identified. The x-intercepts occur when y = 0, while the y-intercept occurs when x = 0, making this distinction a fundamental concept in coordinate geometry.

Final Answer

So, the correct answer is D. (-9,0). We've successfully identified one of the x-intercepts of the quadratic function f(x) = (x-8)(x+9). This demonstrates a clear, step-by-step approach to finding x-intercepts, which involves setting the function equal to zero, using the zero-product property, solving for x, and expressing the solutions as coordinate pairs. This process not only helps in solving similar problems but also builds a strong foundation in understanding quadratic functions and their graphical representations. By breaking down the problem into manageable steps and clearly explaining each stage, we’ve ensured a thorough understanding of how to find x-intercepts. This skill is invaluable in higher-level mathematics and real-world applications, where identifying the roots of equations is a common task.

Key Takeaways

Let's recap the key steps we took to find the x-intercepts:

  1. Set f(x) = 0: This is the fundamental first step because x-intercepts occur where the function's value is zero.
  2. Apply the zero-product property: If a product of factors equals zero, at least one factor must be zero. This allows us to break down the problem into simpler equations.
  3. Solve for x: Find the values of x that make each factor equal to zero.
  4. Express as coordinate pairs: Write the solutions as (x, 0) to represent the points where the graph crosses the x-axis.

Understanding these steps will empower you to tackle a wide range of problems involving quadratic functions and their intercepts. Keep practicing, and you'll master this essential concept in no time! Remember, math is a journey, and every step you take builds your understanding and confidence. You got this!