Hey everyone! Let's dive into a cool problem about rational functions and their y-intercepts. We're going to explore how transformations affect these intercepts and nail down the correct answer. So, grab your thinking caps, and let's get started!
The Essence of Rational Functions
Before we jump into the problem, let's quickly recap what rational functions are all about. In essence, a rational function is a function that can be expressed as the quotient of two polynomials. Think of it as one polynomial divided by another. For example, $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. These functions are fascinating because they can have all sorts of interesting behaviors, like asymptotes and holes, which make them super useful in modeling real-world phenomena. Now, let's break down the critical components that define a rational function. First off, the numerator, which is the polynomial on top, plays a significant role in determining the function's zeros. Zeros are the x-values where the function equals zero, essentially where the graph crosses the x-axis. Next, we have the denominator, the polynomial on the bottom, which is crucial for understanding the function's asymptotes. Asymptotes are lines that the function approaches but never quite touches. They can be vertical, horizontal, or oblique, each giving us valuable insights into the function's behavior as x gets really large or really small. Lastly, the domain of a rational function is all real numbers except for the values that make the denominator zero. These values are excluded because division by zero is undefined. This leads to the presence of vertical asymptotes or holes at those points, adding more character to the function's graph. Understanding these basics is key to tackling more complex problems involving rational functions, including the one we're about to solve!
Delving into the Y-Intercept
The y-intercept of a function is the point where the graph intersects the y-axis. It's a crucial point because it tells us the value of the function when $x = 0$. In simpler terms, it's where our function starts its journey on the coordinate plane, and it often gives us a baseline understanding of the function's behavior. Now, you might wonder, how do we actually find this y-intercept? Well, it’s pretty straightforward! To find the y-intercept, we simply substitute $x = 0$ into the function and solve for $y$. This works because any point on the y-axis has an x-coordinate of 0. So, if we have a function like $f(x) = 2x^2 + 3x + 4$, we find the y-intercept by plugging in $x = 0$: $f(0) = 2(0)^2 + 3(0) + 4 = 4$. Therefore, the y-intercept is the point $(0, 4)$. The y-intercept is more than just a point on a graph; it often carries real-world significance. For example, in a linear function representing the cost of a service, the y-intercept might represent the initial fee or the fixed cost before any service is rendered. In an exponential decay function, the y-intercept could represent the initial amount of a substance that is decaying over time. So, the y-intercept gives us a starting value or a baseline measurement, making it an essential concept in both mathematical analysis and practical applications. When dealing with rational functions, finding the y-intercept is just as straightforward: plug in $x = 0$ into the rational expression and simplify. This simple step can reveal a lot about the function's behavior near the y-axis, making it a fundamental part of our problem-solving toolkit.
The Problem at Hand
Okay, let's bring it back to our specific problem. We're told that we have a rational function $f(x)$ with a y-intercept at $(0, 2)$. This means that when $x = 0$, $f(0) = 2$. Got it? Great! Now, here's the twist: we have a new function $g(x)$ that's defined as $g(x) = 4f(x)$. What this means is that the value of $g(x)$ at any point is four times the value of $f(x)$ at that same point. This is a type of vertical stretch transformation, where the function is stretched away from the x-axis by a factor of 4. So, how does this transformation affect the y-intercept? That's the question we need to answer. To find the y-intercept of $g(x)$, we need to determine the value of $g(0)$. Remember, the y-intercept is where the graph intersects the y-axis, which happens when $x = 0$. We know that $g(x) = 4f(x)$, so to find $g(0)$, we simply substitute $x = 0$ into this equation. This gives us $g(0) = 4f(0)$. Now, we already know that $f(0) = 2$ because the y-intercept of $f(x)$ is $(0, 2)$. So, we can substitute this value into our equation: $g(0) = 4 * 2$. This is where the magic happens! By understanding the relationship between $f(x)$ and $g(x)$, and by using the information about the y-intercept of $f(x)$, we're about to find the y-intercept of $g(x)$. Are you ready for the next step? Let’s calculate that value and solve the problem!
Solving for the New Y-Intercept
Alright, let's crunch those numbers! We've established that $g(0) = 4 * 2$. This is a straightforward calculation, and it tells us that $g(0) = 8$. So, what does this mean in the context of our problem? It means that the y-value of the y-intercept for the function $g(x)$ is 8. Remember, the y-intercept is the point where the graph crosses the y-axis, and it's represented as a coordinate pair $(x, y)$. In this case, the x-coordinate is 0 (since it's on the y-axis), and we've just found that the y-coordinate is 8. Therefore, the y-intercept of the function $g(x)$ is the point $(0, 8)$. This is a direct consequence of the vertical stretch transformation we applied to the original function $f(x)$. By multiplying the entire function by 4, we've essentially stretched the graph vertically, and this has a direct impact on the y-intercept. The original y-intercept of $f(x)$ was $(0, 2)$, and by multiplying the y-value by 4, we've moved the y-intercept to $(0, 8)$. This illustrates a fundamental principle of function transformations: vertical stretches and compressions affect the y-values of points on the graph, including the y-intercept. It's a powerful concept that allows us to predict how transformations will change the key features of a function. Now that we've found the y-intercept of $g(x)$, we've essentially solved the problem. We took the given information, applied the principles of function transformations, and arrived at the correct answer. But let's take a moment to reflect on what we've learned and how we can apply these concepts to other problems.
Final Answer and Recap
So, to wrap things up, the point that represents the y-intercept of the function $g(x)$ is $(0, 8)$. Woohoo! We nailed it! Let's quickly recap the journey we took to get here. We started with a rational function $f(x)$ that had a y-intercept at $(0, 2)$. This gave us the crucial piece of information that $f(0) = 2$. Then, we were introduced to a new function $g(x)$ defined as $g(x) = 4f(x)$. This meant that $g(x)$ is a vertical stretch of $f(x)$ by a factor of 4. To find the y-intercept of $g(x)$, we needed to find $g(0)$. By substituting $x = 0$ into the equation for $g(x)$, we got $g(0) = 4f(0)$. Since we knew $f(0) = 2$, we could substitute this value to get $g(0) = 4 * 2 = 8$. Therefore, the y-intercept of $g(x)$ is $(0, 8)$. This problem beautifully illustrates how transformations of functions affect their key features, such as the y-intercept. A vertical stretch by a factor of 4 simply multiplies the y-value of the y-intercept by 4. This is a powerful concept that applies not only to rational functions but to all types of functions. By understanding these transformations, we can quickly analyze and predict the behavior of functions, making problem-solving much more efficient. Guys, remember that the key to mastering these concepts is practice. Try working through similar problems, experimenting with different transformations, and visualizing how they affect the graph of the function. With a little bit of effort, you'll become a pro at function transformations in no time! Keep up the great work, and I'll see you in the next problem!