Finding X-Intercepts Of Dilated Functions

Hey math enthusiasts! Let's dive into a fun problem involving functions, specifically focusing on how dilations impact their x-intercepts. We'll be looking at the function f(x) = (x + 5)(x - 3) and how it transforms when dilated by x to create a new function, g(x) = x * f(x). Our goal? To figure out how many x-intercepts g(x) has. Sounds good? Let's get started!

Understanding the Basics: Functions and X-Intercepts

Okay, before we jump into the nitty-gritty, let's refresh our memories on some key concepts. In the world of functions, an x-intercept is simply the point where the graph of the function crosses or touches the x-axis. At this point, the value of y (or f(x), g(x), etc.) is always zero. Think of it as the spot where the function 'hits' zero. Finding these x-intercepts is crucial in understanding the behavior of a function because they tell us where the function's value is neither positive nor negative. It's like finding the function's 'roots' or 'zeros.'

Now, let's talk about the original function, f(x) = (x + 5)(x - 3). This function is a quadratic function, meaning its graph is a parabola. To find the x-intercepts of f(x), we need to solve the equation (x + 5)(x - 3) = 0. This is where things get super simple. Because of the zero-product property, which states that if the product of factors is zero, at least one of the factors must be zero. We set each factor equal to zero and solve. So, we have:

• x + 5 = 0, which gives us x = -5
• x - 3 = 0, which gives us x = 3

Therefore, f(x) has two x-intercepts: x = -5 and x = 3. These are the points where the parabola crosses the x-axis. Remember that these intercepts are the solutions to the equation f(x) = 0. Always keep in mind that the x-intercept is where the graph crosses the x-axis, making the y-value (or function value) equal to zero. These concepts are fundamental to grasping the problem at hand.

Now, let's move on to the interesting part: dilation. What happens to these intercepts when we change the function?

The Impact of Dilation: Transforming the Function

Here's where the problem gets exciting! We're not just dealing with the original function f(x) anymore. Instead, we're introducing a dilation by x. What does this mean? It means we're multiplying the entire function f(x) by x. This creates a new function, g(x) = x * f(x). So, let's substitute f(x):

• g(x) = x * (x + 5)(x - 3)

This transformation changes the shape and behavior of the function's graph. Dilation stretches or compresses the graph, affecting its intercepts. In this case, multiplying by x significantly changes the x-intercepts and the overall appearance of the graph. The original intercepts of f(x) are still important, but now we have to consider how the x factor impacts the new function. Understanding how dilation affects the x-intercepts is key to solving our problem. So, let's find the x-intercepts of the dilated function, g(x).

To find the x-intercepts of g(x), we need to solve the equation g(x) = 0. So we set x * (x + 5)(x - 3) = 0. Again, applying the zero-product property, we look for values of x that make the equation true. We have three factors that could equal zero:

• x = 0
• x + 5 = 0, which gives us x = -5
• x - 3 = 0, which gives us x = 3

So, the x-intercepts of g(x) are x = 0, x = -5, and x = 3. This means the graph of g(x) crosses the x-axis at these three points. We started with two x-intercepts and now have three! The dilation by x has added a new x-intercept at x = 0. This is a crucial observation.

Unveiling the Answer: Counting the X-Intercepts

Alright, we've done the calculations, and we're ready to answer the big question: How many x-intercepts does g(x) have? From our calculations, we found that g(x) has three x-intercepts: x = 0, x = -5, and x = 3. So the correct answer is three!

Here's a quick recap of our journey:

1. Understanding X-Intercepts: We started by understanding what x-intercepts are: the points where a function crosses the x-axis.
2. Original Function: We looked at the original function, f(x) = (x + 5)(x - 3), and found its two x-intercepts at x = -5 and x = 3.
3. Dilation: We dilated f(x) by x to get g(x) = x * (x + 5)(x - 3).
4. Finding New Intercepts: We found the x-intercepts of g(x) by setting it equal to zero and solving. This gave us three x-intercepts: x = 0, x = -5, and x = 3.
5. The Answer: g(x) has three x-intercepts.

The dilation by x introduced a new intercept at x = 0, changing the number of x-intercepts. This is a common phenomenon in function transformations, and understanding how these transformations impact key features like x-intercepts is vital.

Visualizing the Transformation: A Graphical Perspective

To really solidify our understanding, let's take a moment to visualize what's happening graphically. You can use graphing tools like Desmos or Wolfram Alpha to plot both f(x) and g(x). When you graph f(x) = (x + 5)(x - 3), you'll see a parabola intersecting the x-axis at x = -5 and x = 3. This is a standard upward-facing parabola. Now, when you graph g(x) = x * (x + 5)(x - 3), you'll see a cubic function. The key here is the addition of the factor x, which changes the shape of the graph dramatically. The cubic function now intersects the x-axis at three points: x = 0, x = -5, and x = 3.

This visual representation helps cement the concept. You can literally 'see' how the dilation by x shifts the graph and creates that new x-intercept at x = 0. You can also see how the original x-intercepts are retained. It's always beneficial to confirm your mathematical reasoning with a visual tool. Understanding the graphical interpretation adds another layer of understanding to the problem. It is much easier to digest the information when we can see it with our eyes, and this is why graphing tools are so powerful. Moreover, practicing with different functions and dilations will improve your understanding of how functions transform and how their properties change as a result.

Why This Matters: Real-World Applications

You might be wondering, why does any of this matter? Well, understanding function transformations and x-intercepts has applications in many areas, including:

• Physics: Modeling projectile motion, where x-intercepts can represent when an object hits the ground.
• Engineering: Designing structures and systems, where understanding how functions behave is critical.
• Economics: Analyzing supply and demand curves, where intercepts can reveal key points like equilibrium.
• Computer Graphics: Creating realistic images and animations.

Function transformations are a fundamental concept in mathematics and have widespread applications in various fields. From modeling physical phenomena to analyzing data, understanding function transformations like dilation by x provides a powerful toolset for problem-solving. It helps to understand how the changes in one part of a system affect the whole system.

Conclusion: Mastering Function Transformations

So, there you have it! We've successfully navigated the world of function transformations and x-intercepts. We learned how dilation by x changes the number of x-intercepts. The key takeaway is to carefully analyze the function, consider the impact of transformations, and utilize the zero-product property. Always remember to break down the problem into smaller steps. Understand the basics, and then apply those principles to the more complex scenarios. Keep practicing, exploring, and most importantly, have fun with math!

This problem offers a great introduction to the concepts of function transformation and the zero-product property, which are valuable tools in your mathematical arsenal. Keep exploring, and you'll find that math can be both challenging and incredibly rewarding.

For further exploration, you might find the following resources helpful:

• Khan Academy (https://www.khanacademy.org/): Khan Academy has excellent videos and practice exercises on function transformations and solving quadratic equations. This is a great place to start if you want to further study the concepts discussed.

Keep practicing, and you'll become a function transformation master in no time!