Reflecting Functions: Find G(x) Across The X-Axis

Alright guys, let's dive into a fun little problem where we're going to reflect a function across the x-axis. It's like looking in a mirror, but with math! We'll start with a function, $f(x)$, and then find its reflection, $g(x)$, over the x-axis. This involves a neat little trick that makes the whole process pretty straightforward. So, grab your pencils, and let's get started!

Understanding Reflections Across the X-Axis

Before we jump into the specifics, let's quickly recap what it means to reflect a function across the x-axis. Imagine you have a graph of a function plotted on a coordinate plane. The x-axis is that horizontal line running right through the middle. To reflect the function across the x-axis, you're essentially flipping the graph upside down. Any point $(x, y)$ on the original graph will become $(x, -y)$ on the reflected graph. In simpler terms, the x-value stays the same, but the y-value changes its sign. This is the key concept we'll use to find our reflected function $g(x)$. Remember, reflecting across the x-axis means every y-value becomes its opposite, so positive y-values become negative, and negative y-values become positive. Zero y-values stay at zero, as they are already on the x-axis. Got it? Great, let's move on to the problem at hand.

Reflecting a function across the x-axis involves changing the sign of the function's output. Specifically, if you have a function f(x), its reflection across the x-axis, denoted as g(x), is given by g(x) = -f(x). This transformation essentially flips the graph of the function over the x-axis. Any point (x, y) on the graph of f(x) becomes (x, -y) on the graph of g(x). For instance, if f(2) = 3, then g(2) = -3. The x-coordinate remains the same, while the y-coordinate changes its sign. This concept is fundamental in understanding how functions behave under reflections. Thinking about it graphically can provide a clear intuition: imagine the x-axis as a mirror; the reflection of the function appears as its mirror image. This simple but powerful idea allows us to easily find the reflected function by just negating the original function's expression. Understanding reflections is crucial in various areas of mathematics, including geometry, calculus, and linear algebra.

The Given Function: $f(x) = 9x - 3$

Okay, so we're given the function $f(x) = 9x - 3$. This is a linear function, which means when we graph it, we'll get a straight line. The '9' in front of the 'x' tells us the slope of the line (how steep it is), and the '-3' tells us where the line crosses the y-axis (the y-intercept). Now, to find the reflection of this function across the x-axis, we need to apply the concept we just discussed. Remember, reflecting across the x-axis means we need to change the sign of the entire function. In other words, we need to find $g(x)$ such that $g(x) = -f(x)$. This is a crucial step, so make sure you understand why we're doing this. We are essentially flipping every point on the line $f(x)$ over the x-axis to get the new line $g(x)$. This transformation will affect both the slope and the y-intercept of the original line. So, let's proceed with the calculation and see what we get.

Understanding the Original Function

The function $f(x) = 9x - 3$ is a linear equation. It represents a straight line on a coordinate plane. The '9' is the slope, indicating how steeply the line rises or falls. A positive slope means the line goes upwards as you move from left to right. The '-3' is the y-intercept, which is the point where the line crosses the y-axis. This point is (0, -3). To graph this line, you could plot a couple of points and draw a line through them. For example, when x = 0, y = -3, and when x = 1, y = 6. Connecting these points gives you the line represented by f(x) = 9x - 3. Understanding these components helps visualize the function and predict how it will change when reflected.

Finding the Reflection: $g(x) = -f(x)$

Here comes the fun part! We know that $g(x) = -f(x)$, and we know that $f(x) = 9x - 3$. So, to find $g(x)$, we simply need to multiply the entire expression for $f(x)$ by -1. This means we have:

$g(x) = -(9x - 3)$

Now, we distribute the negative sign to both terms inside the parentheses:

$g(x) = -9x + 3$

And there you have it! The reflection of $f(x) = 9x - 3$ across the x-axis is $g(x) = -9x + 3$. Notice how the sign of both the slope and the y-intercept changed. The original function had a positive slope of 9 and a y-intercept of -3. The reflected function has a negative slope of -9 and a y-intercept of +3. This makes sense because reflecting across the x-axis flips the line upside down, changing the direction of the slope and the position of the y-intercept relative to the x-axis.

Detailed Calculation

To find g(x), we substitute f(x) into the equation g(x) = -f(x):

g(x) = -(9x - 3)

Next, we distribute the negative sign across the terms inside the parentheses:

g(x) = -9x - (-3)

Simplifying, we get:

g(x) = -9x + 3

This is the equation of the reflected function. The slope has changed from 9 to -9, and the y-intercept has changed from -3 to 3. This confirms that our reflection is correct. Always double-check your work to ensure the signs are correct and the distribution is accurate. This simple algebraic manipulation gives us the reflected function, which is a key step in understanding transformations of functions.

The Answer in the Form $mx + b$

We've found that $g(x) = -9x + 3$. Now, we need to express our answer in the form $mx + b$, where $m$ and $b$ are integers. Looking at our equation, we can see that it's already in the correct form. Here, $m = -9$ and $b = 3$. Both -9 and 3 are integers, so we're good to go! This is the final step to ensure we've answered the question completely and accurately. Always make sure your answer is in the requested format to avoid losing points. In this case, we've clearly identified the values of $m$ and $b$, confirming that our solution is correct and complete. High five, guys! We did it.

Verification

To verify our result, we can pick a point on the original function and see where it ends up on the reflected function. For example, let's take x = 1. On the original function:

f(1) = 9(1) - 3 = 6

So the point (1, 6) is on the graph of f(x). Now let's check the reflected function:

g(1) = -9(1) + 3 = -6

So the point (1, -6) is on the graph of g(x). Notice that the x-value stayed the same, but the y-value changed from 6 to -6. This confirms that we have indeed reflected the function across the x-axis. This simple check can help ensure that your answer is correct and you understand the transformation.

Final Answer

Therefore, the reflection of $f(x) = 9x - 3$ across the x-axis is $g(x) = -9x + 3$. In the form $mx + b$, we have $m = -9$ and $b = 3$. And that, my friends, is how you reflect a function across the x-axis! Hopefully, this explanation has been clear and helpful. Remember, the key is to change the sign of the entire function. Keep practicing, and you'll become a pro at these transformations in no time! Great job, everyone!

In summary:

• Original Function: f(x) = 9x - 3
• Reflection across the x-axis: g(x) = -9x + 3
• Form mx + b: m = -9, b = 3

With this, we have successfully found the reflected function and expressed it in the desired form. Keep up the excellent work! Remember to always double-check your signs and calculations, and you'll be golden. Now go forth and conquer more math problems!