Function Reflections: F(x) And G(x) Explained

Hey math enthusiasts! Let's dive into the fascinating world of functions and explore the relationship between $f(x) = 0.7(6)^x$ and $g(x) = 0.7(6)^{-x}$. This is a classic example of how a simple change in the equation can dramatically alter the graph of a function. We're going to break down their connection, exploring the concept of reflections and how they impact these exponential functions. No need to worry if you're feeling a little rusty – we'll go through it step by step, making sure you grasp the core concepts. This is like a fun journey into the visual side of math, where we get to see how equations translate into shapes and how those shapes relate to each other. Ready to unravel the mystery? Let's get started!

Understanding Exponential Functions and Their Graphs

Before we jump into reflections, let's quickly recap what exponential functions are all about. In general, an exponential function takes the form $f(x) = a imes b^x$, where: $a$ is the initial value (the value of the function when $x = 0$); $b$ is the base, which determines the rate of growth or decay. If $b > 1$, the function increases (grows) as $x$ increases. If $0 < b < 1$, the function decreases (decays) as $x$ increases. The graphs of exponential functions have a characteristic curve. They either shoot upwards rapidly (for growth functions) or flatten out towards the x-axis (for decay functions). The constant 'a' just stretches or compresses the graph vertically. In our specific case, $f(x) = 0.7(6)^x$, we can see that: The initial value is $0.7$; The base is $6$, which is greater than $1$, so this is an exponential growth function. This means as $x$ gets larger, $f(x)$ increases rapidly. The graph starts close to the x-axis (because of the 0.7) and curves upwards. Understanding this basic structure is crucial for understanding how reflections change the graph.

The Basics of Graphing

When graphing, remember that the x-axis is the horizontal line and the y-axis is the vertical line. Each point on the graph is represented by an (x, y) coordinate. The x-coordinate tells you how far to move horizontally from the origin (0,0), and the y-coordinate tells you how far to move vertically. For the function $f(x) = 0.7(6)^x$, as $x$ increases, $f(x)$ grows exponentially. If you plug in $x = 0$, you get $f(0) = 0.7(6)^0 = 0.7$. This means the graph passes through the point (0, 0.7). If you plug in $x = 1$, you get $f(1) = 0.7(6)^1 = 4.2$. So the graph passes through (1, 4.2). The curve increases rapidly. As x becomes negative, $f(x)$ gets smaller but never touches the x-axis (it approaches it, but never crosses it). Now, let's explore what happens when we change the equation to $g(x) = 0.7(6)^{-x}$.

The Role of the y-axis in Reflections

Now, let's focus on the key concept: reflections. A reflection is like looking at the graph in a mirror. When we talk about reflecting a graph over the y-axis, imagine the y-axis is the mirror. Every point on the original graph ($f(x)$) has a corresponding point on the reflected graph ($g(x)$) that is the same distance from the y-axis but on the opposite side. To reflect a function over the y-axis, you change the sign of the x in the equation. In our example, the original function is $f(x) = 0.7(6)^x$. The reflected function over the y-axis is $g(x) = f(-x) = 0.7(6)^{-x}$. See how the only difference is the negative sign in the exponent? This simple change causes a horizontal reflection. The shape of the graph remains the same, but it's flipped over the y-axis. The y-intercept (where the graph crosses the y-axis) stays the same, but the rest of the points flip. For example, if $f(1) = 4.2$, then $g(-1) = 4.2$. This transformation gives the function $g(x)$ a decreasing trend as $x$ increases, unlike $f(x)$, which increases as $x$ increases. The two functions are mirror images of each other with respect to the y-axis. This is the heart of understanding the relationship between the two functions.

Practical Implications of the Reflection

Understanding reflections is more than just a theoretical concept. It helps you quickly visualize and analyze functions. For instance, if you are given the graph of $f(x) = 0.7(6)^x$, you instantly know what the graph of $g(x) = 0.7(6)^{-x}$ looks like. It is the same curve, flipped across the y-axis. This is useful in various fields, such as physics and engineering, where reflections are essential. In simple terms, think of it like this: If $f(x)$ represents the growth of something, then $g(x)$ represents the decay of something, which is exactly the mirror image of that growth. This understanding allows you to predict the behavior of systems by manipulating their mathematical representations. It also helps in quickly identifying key properties of functions like the intercepts and asymptotes (lines that the graph approaches but never touches). In our case, the y-intercept of both functions is the same (0, 0.7), and the x-axis is a horizontal asymptote. The only difference is the direction of the curve.

The Answer: Reflecting Over the y-axis

So, to answer the question, the correct relationship between $f(x) = 0.7(6)^x$ and $g(x) = 0.7(6)^{-x}$ is: B. $g(x)$ is the reflection of $f(x)$ over the $y$-axis. The negative sign in the exponent of $g(x)$ is the key. It causes a horizontal flip. Remember that reflecting over the x-axis would mean changing the sign of the entire function (multiplying the whole equation by -1), which is not the case here. This creates a vertical flip. And, reflecting over both axes is equivalent to a rotation of 180 degrees around the origin, which would also change the equation differently. So, the correct answer is indeed the y-axis reflection. Congratulations, you've successfully navigated the world of exponential functions and reflections!

Quick Recap

Here’s a quick summary to cement your understanding:

• Exponential functions have the form $f(x) = a imes b^x$.
• The base b determines whether the function grows or decays.
• Reflecting a function over the y-axis involves changing the sign of the x in the equation.
• $f(x) = 0.7(6)^x$ is an exponential growth function.
• $g(x) = 0.7(6)^{-x}$ is the reflection of $f(x)$ over the y-axis, making it an exponential decay function.
• Reflections are a fundamental concept in understanding and manipulating functions.

Keep practicing, and you'll become a pro at recognizing and understanding these transformations. Keep up the great work, and don't hesitate to explore more examples and functions. Math is an exciting journey and with each step, your knowledge expands! Happy learning!