Function Translation: Finding The Image Of F(x)

Hey guys! Ever wondered how a function changes when you slide it around the coordinate plane? That's what we call translation, and it's a super important concept in math, especially when dealing with functions. Today, we're going to dive deep into translating a specific function and finding its new image. We'll break it down step by step, so even if you're new to this, you'll be a pro in no time!

Understanding Function Translation

Before we jump into the problem, let's get a solid grasp on what function translation actually means. Imagine you have a graph of a function. Translating it is like picking it up and moving it to a new spot without changing its shape or orientation. Think of it like sliding a sticker on a page – the sticker itself stays the same, but its position changes.

In mathematical terms, a translation is defined by a vector. A vector tells us how far to move the function horizontally and vertically. For example, the vector $\begin{pmatrix} a \\ b \end{pmatrix}$ means we shift the function 'a' units along the x-axis and 'b' units along the y-axis. If 'a' is positive, we move to the right; if it's negative, we move to the left. Similarly, if 'b' is positive, we move upwards; if it's negative, we move downwards.

So, when we translate a function f(x) by a vector $\begin{pmatrix} a \\ b \end{pmatrix}$, we're essentially creating a new function, let's call it g(x), which is a shifted version of f(x). The relationship between f(x) and g(x) is the key to solving these kinds of problems.

To really nail this down, let's consider a simple example. Suppose we have the function f(x) = x². This is a parabola, a U-shaped curve. Now, let's translate it by the vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$. This means we're shifting the parabola 2 units to the right and 3 units upwards. The new function, g(x), will still be a parabola, but its vertex (the bottom point of the U) will be in a different location. Understanding how the vector affects the function's equation is crucial, and we'll see how that works in the main problem we're tackling today. Keep this analogy of sliding a graph in your mind, and you'll be well on your way to mastering function translations.

The Problem: Translating f(x) = (2x + 3) / (x - 5)

Alright, let's tackle the problem at hand. We're given the function f(x) = (2x + 3) / (x - 5) and we want to find its image after a translation by the vector $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$. This looks a bit more complex than a simple parabola, but don't worry, we'll break it down into manageable steps.

First, let's recap what this vector means. The vector $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$ tells us we need to shift the function 5 units to the right (along the x-axis) and 7 units upwards (along the y-axis). The core idea here is to figure out how these shifts affect the function's equation. Remember, translating a function means changing its position on the graph, but the fundamental relationship between x and y values remains the same, just in a new location.

To find the image of the function after the translation, we need to find a new function, let's call it g(x), that represents the translated graph. Here's the key concept: if a point (x, y) lies on the graph of f(x), then the point (x + 5, y + 7) will lie on the graph of g(x). This is because we've shifted every point on the original graph 5 units to the right and 7 units upwards.

Now, let's translate this geometric idea into algebra. If we let (x', y') be a point on the translated function g(x), then we know that x' = x + 5 and y' = y + 7, where (x, y) is the corresponding point on the original function f(x). Our goal is to express y' in terms of x', which will give us the equation for g(x). To do this, we need to reverse the transformations to express x and y in terms of x' and y'. This is a crucial step in finding the translated function. Once we have x and y in terms of the new coordinates, we can substitute them into the original function's equation and see what we get!

Step-by-Step Solution

Okay, let's get our hands dirty and solve this problem step by step. This is where the magic happens, guys! We'll take the concepts we've discussed and put them into action.

Step 1: Express the original coordinates in terms of the translated coordinates.

As we discussed earlier, the translation vector $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$ tells us that x' = x + 5 and y' = y + 7, where (x', y') is a point on the translated function g(x) and (x, y) is the corresponding point on the original function f(x). To express x and y in terms of x' and y', we simply rearrange these equations:

• x = x' - 5
• y = y' - 7

This is a crucial step because we need to substitute these expressions into the original function's equation. We're essentially undoing the translation to see how the original function's variables relate to the new translated variables. Think of it like rewinding a movie – we're going back to the original coordinates.

Step 2: Substitute into the original function's equation.

Now, we know that y = f(x), and we have the equation for f(x): f(x) = (2x + 3) / (x - 5). We also have expressions for x and y in terms of x' and y'. Let's substitute them into the equation:

y' - 7 = (2(x' - 5) + 3) / ((x' - 5) - 5)

This might look a bit messy, but don't panic! We're just plugging in the expressions we found in Step 1. The next step is to simplify this equation and isolate y', which will give us the equation for our translated function, g(x'). Remember, g(x') is just a way of saying the translated function in terms of the new x-coordinate, x'. This substitution is the heart of the solution, guys – it's where we connect the original function to its translated image.

Step 3: Simplify the equation and isolate y'.

Let's simplify the equation we obtained in Step 2. This involves some algebraic manipulation, but nothing too scary. We'll just take it one step at a time. First, let's expand the terms in the numerator and the denominator:

y' - 7 = (2x' - 10 + 3) / (x' - 5 - 5)

Now, let's combine the constants:

y' - 7 = (2x' - 7) / (x' - 10)

Finally, to isolate y', we add 7 to both sides of the equation:

y' = (2x' - 7) / (x' - 10) + 7

To combine the terms on the right side, we need a common denominator. We'll write 7 as 7*(x' - 10) / (x' - 10):

y' = (2x' - 7) / (x' - 10) + 7(x' - 10) / (x' - 10)

Now we can add the numerators:

y' = (2x' - 7 + 7x' - 70) / (x' - 10)

Combine like terms in the numerator:

y' = (9x' - 77) / (x' - 10)

Step 4: Write the equation of the translated function g(x).

We've done it! We've simplified the equation and isolated y'. Now we have the equation for the translated function. Remember, y' is just g(x'), so we can write:

g(x') = (9x' - 77) / (x' - 10)

To make it look a bit cleaner, we can replace x' with x (since x' is just a dummy variable representing the input of the translated function):

g(x) = (9x - 77) / (x - 10)

This is the image of the function f(x) = (2x + 3) / (x - 5) after being translated by the vector $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$. Woohoo! We found the translated function, and it wasn't so bad after all, right? Remember the key steps: express the original coordinates in terms of the translated coordinates, substitute into the original equation, simplify, and you've got it!

Conclusion

So, there you have it! We've successfully found the image of the function f(x) = (2x + 3) / (x - 5) after a translation by the vector $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$. The translated function is g(x) = (9x - 77) / (x - 10). Guys, this might seem like a complex problem at first, but by breaking it down into smaller steps, we can conquer it!

The key takeaway here is understanding how translations affect the coordinates of points on a graph and how to translate that geometric understanding into algebraic manipulations. Remember the steps we followed: expressing the original coordinates in terms of the translated coordinates, substituting into the original function's equation, and simplifying to find the equation of the translated function. These steps are applicable to translating various types of functions, not just rational functions like the one we worked with today.

Function translation is a fundamental concept in mathematics and has applications in various fields, such as computer graphics, physics, and engineering. Understanding how to manipulate functions through transformations like translation is crucial for further studies in mathematics and related disciplines. So keep practicing, and you'll become a master of function transformations in no time! Keep up the awesome work, everyone!