Determining The Image Of Function F(x) = 2x+1 After Translation

Understanding Translations in Mathematics

In the vast world of mathematics, transformations play a crucial role in understanding how geometric figures and functions can be manipulated in space. Among these transformations, translation stands out as a fundamental concept. Guys, think of translation as a simple slide – moving a figure or function without rotating or resizing it. It's like shifting a piece on a chessboard; the piece remains the same, but its position changes.

In our specific problem, we're dealing with a linear function, F(x) = 2x + 1, and we want to see what happens when we translate it. But what does it mean to translate a function? Well, imagine the graph of the function as a line on a coordinate plane. Translating the function means shifting this entire line to a new position on the plane. This shift is defined by a translation vector, which tells us how far to move the line horizontally and vertically. In this case, our translation vector is (3, -2). This means we're going to slide the line 3 units to the right and 2 units down. So, the keyword here is translation, which involves moving a function or shape without changing its size or orientation. The translation vector (3, -2) tells us exactly how to shift the function F(x) = 2x + 1. Let's dive into the math behind this and see how we can find the new function after this translation.

The Mathematics Behind Function Translation

To truly understand how function translation works, let's break down the mathematical principles involved. When we translate a function F(x) using a translation vector (a, b), we're essentially creating a new function, let's call it G(x). The relationship between F(x) and G(x) is defined by how the input and output values change after the translation. Guys, think about it this way: if we want to find the value of G(x) at a particular point, we need to figure out what the corresponding point was in the original function F(x) before the translation. The horizontal component 'a' of the translation vector affects the input 'x', while the vertical component 'b' affects the output F(x). Specifically, to find G(x), we need to evaluate F(x - a) and then add 'b' to the result. This might sound a bit complicated, but it's actually quite intuitive when you visualize it. The key idea is that we're undoing the horizontal shift in the input and then applying the vertical shift to the output. In our problem, the translation vector is (3, -2), so 'a' is 3 and 'b' is -2. This means we need to find F(x - 3) and then subtract 2 from it. This process is the core of understanding how translations alter functions, and it's essential for solving problems like this one. Remember, the translation vector dictates the shift, and we use it to adjust both the input and output of the original function. So, let's see how this applies to our specific function, F(x) = 2x + 1.

Applying the Translation to F(x) = 2x + 1

Now, let's get our hands dirty and apply the translation (3, -2) to the function F(x) = 2x + 1. Remember, the formula for translating a function is G(x) = F(x - a) + b, where (a, b) is the translation vector. In our case, a = 3 and b = -2. So, the first step is to find F(x - 3). To do this, we replace every 'x' in the original function with '(x - 3)'. So, F(x - 3) becomes 2(x - 3) + 1. Now, let's simplify this expression: 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5. Great! We've found F(x - 3). But we're not done yet. We still need to add 'b' to this result. Remember, b = -2. So, G(x) = F(x - 3) + b = (2x - 5) + (-2) = 2x - 5 - 2 = 2x - 7. There you have it! The translated function, G(x), is 2x - 7. This means that if we were to graph both F(x) = 2x + 1 and G(x) = 2x - 7, we would see that G(x) is simply F(x) shifted 3 units to the right and 2 units down. Guys, isn't it cool how math lets us visualize these transformations so clearly? We took the original function, applied the translation vector, and found the new function with ease. So, the key steps here were substituting (x - 3) into the original function, simplifying the expression, and then adding -2 to account for the vertical shift. Let's summarize our findings and make sure we've got the final answer.

The Final Image: G(x) = 2x - 7

Alright, guys, let's recap what we've done and nail down the final answer. We started with the function F(x) = 2x + 1 and wanted to find its image after a translation of (3, -2). We understood that translation means shifting the function without changing its shape or size. Using the formula G(x) = F(x - a) + b, where (a, b) is the translation vector, we systematically worked through the problem. We substituted (x - 3) into F(x), simplified the expression, and then added -2 to account for the vertical shift. After all the calculations, we arrived at the translated function: G(x) = 2x - 7. This is the image of the function F(x) = 2x + 1 after the translation. What does this mean in simple terms? It means that every point on the graph of F(x) has been moved 3 units to the right and 2 units down to create the graph of G(x). The slope of the line remains the same (which is 2), but the y-intercept has changed (from 1 to -7). This visually confirms that we've indeed performed a translation. So, the final answer to the question is G(x) = 2x - 7. This showcases how powerful mathematical transformations can be in manipulating functions and understanding their behavior. By applying the translation vector correctly, we've successfully found the image of the function. This process is crucial in various fields, from computer graphics to physics, where understanding transformations is essential.

Visualizing the Translation (Optional)

For those of you who are visual learners, it can be incredibly helpful to actually see what this translation looks like. Imagine the graph of F(x) = 2x + 1. It's a straight line with a slope of 2 and a y-intercept of 1. Now, picture shifting this entire line 3 units to the right and 2 units down. What do you get? You get another straight line, G(x) = 2x - 7, which also has a slope of 2 but a different y-intercept of -7. Guys, you can even plot these two lines on a graph to see the translation in action. Use a graphing calculator or an online tool like Desmos to visualize the functions. You'll clearly see how the original line has been moved to a new position on the coordinate plane. This visual representation reinforces the concept of translation and makes it even easier to understand. Seeing the transformation happening graphically can solidify your understanding and make it easier to tackle similar problems in the future. Visualizing mathematical concepts is a powerful technique, and it can often provide insights that equations alone might not reveal. So, if you're ever struggling to grasp a mathematical idea, try to visualize it – you might be surprised at how much it helps!