Unveiling Transformations: Translation And Dilation In Coordinate Geometry

Hey guys! Let's dive into the fascinating world of coordinate geometry, specifically focusing on transformations. We'll explore how lines change when we shift them around (translation) and how they expand or contract (dilation). Buckle up, because we're about to make some math magic!

2. Unveiling the Translated Line: Understanding the Basics of Translation

Alright, let's tackle the first problem: finding the equation of the line after it's been translated. The initial equation of the line is y = 2x + 3, and it is translated by the vector (3, 2). Now, what does this translation thing actually mean? Think of it like this: every single point on the line gets shifted horizontally by 3 units and vertically by 2 units. It's like taking the entire line and giving it a little nudge in a specific direction. The essence of the transformation lies in how we deal with the coordinate changes. To find the equation of the transformed line, we need to express the transformation algebraically.

Let's assume a general point on the original line is (x, y). After the translation, this point becomes (x', y'). The translation vector (3, 2) tells us how the coordinates change: x' = x + 3 and y' = y + 2. However, our main goal is to find the equation of the new line in terms of x and y. So, we need to express x and y in terms of x' and y'. From the equations above, we can rearrange them to get: x = x' - 3 and y = y' - 2. These equations are crucial because they tell us how to substitute the original coordinates in terms of the new coordinates. Then, we substitute these back into the original equation of the line y = 2x + 3. Replacing 'y' with 'y' - 2 and 'x' with 'x' - 3, we get y' - 2 = 2(x' - 3) + 3. Then, it's just a matter of simplifying this equation. Expanding the right side of the equation, we get y' - 2 = 2x' - 6 + 3. Combine the constant terms: y' - 2 = 2x' - 3. Finally, we can isolate y' by adding 2 to both sides of the equation, to get the equation of the transformed line: y' = 2x' - 1. Remember, the primes ( ' ) just indicate that these are the new coordinates after the translation. We can drop these primes and rewrite the equation as y = 2x - 1. So, the equation of the translated line is y = 2x - 1. This means our translated line is parallel to the original line, and shifted downward. This is how we find the equation of a line after translation. Understanding these steps and concepts is super important for doing well in coordinate geometry.

Practical Application: Visualizing the Transformation

To really get a feel for this, imagine plotting both the original line y = 2x + 3 and the transformed line y = 2x - 1 on a graph. The lines would be parallel, and the second line (the translated one) would be located below the first one. This visualization is important because it confirms our calculations. Also, it allows us to visualize what is going on with the coordinate system. You can even pick a couple of points on the original line, apply the translation vector, and see that they land on the translated line. This is a neat way to check your work and build your understanding. Also, note that the slope of the line does not change in the transformation since it is only a translation. Only the y-intercept is altered. By following these steps and visualizing the transformation, you'll be able to confidently solve any translation problems thrown your way!

3. Delving into Dilation: Scaling Lines in Coordinate Geometry

Okay, let's explore dilation. Dilation is a transformation that changes the size of a shape. We are given the line 5x + 2y - 7 = 0 and are told it's dilated by a scale factor of k = 3, with the center of dilation at a specific point. Dilation with a scale factor of 3 means the line is going to be scaled up; every distance from the center of dilation is going to be multiplied by 3. Also, it's super important to note how the center of dilation affects the final transformed equation. But before we get to the final answer, let's understand the process of dilation, its formulas, and how to apply them.

Before we dive in, let’s quickly discuss the concept of dilation and its formula. If we have a point (x, y) and dilate it with respect to a center (a, b) with a scale factor of k, the new coordinates (x', y') are given by the following formulas: x' = a + k(x - a) and y' = b + k(y - b). Let's go through the steps needed to find the dilated equation. Since we are given the equation of the line, and are asked for the dilated line equation, we need to know the center of dilation, which is (a, b). Let's assume the center of dilation is the origin, (0, 0), just to get started. Given the scale factor k = 3, we can apply the dilation formula to the general point on the line (x, y). Remember our formulas for dilation? x' = a + k(x - a) and y' = b + k(y - b). Since the center of dilation is (0, 0), the formulas simplify to x' = 0 + 3(x - 0) = 3x and y' = 0 + 3(y - 0) = 3y. The equations for the coordinates can be rewritten as x = x'/3 and y = y'/3. These equations are our key for substituting in the original line equation. Then, we substitute these back into the original equation of the line 5x + 2y - 7 = 0. Replacing 'x' with 'x'/3 and 'y' with 'y'/3, we get 5(x'/3) + 2(y'/3) - 7 = 0. Simplify and solve the equation. The next step is to clear the fractions. Multiply the entire equation by 3: 5x' + 2y' - 21 = 0. Then, it's just a matter of rewriting this, so you can drop the primes and rewrite it as 5x + 2y - 21 = 0. This is the equation of the dilated line. This transformation has the same slope as the original line and a different y-intercept.

The Role of the Center of Dilation

The choice of the center of dilation dramatically affects the final result. If the center of dilation is at the origin, the process is pretty straightforward, as we have already seen. However, if the center is at another point, the transformations become more complex. Remember the formulas: x' = a + k(x - a) and y' = b + k(y - b), where (a, b) is the center. So, if the center of dilation were, for example, the point (1, 2), the transformation would be very different. The line would still be scaled by a factor of 3, but the position of the new line would be very different.

Final Thoughts on Dilation

Dilation changes the size of a line, but it preserves its direction if the dilation center is not on the line. The slope does not change, just the intercept. This concept is fundamental in coordinate geometry and provides the basis for understanding more advanced concepts like similar figures. So, make sure you understand the formulas and practice with different examples to solidify your understanding.

Conclusion: Mastering Transformations

Awesome work, guys! We've successfully navigated the worlds of translation and dilation, and now you have a better understanding of how lines change with each transformation. Keep practicing and exploring these concepts. Coordinate geometry is fun, and with a little effort, you'll be acing these problems in no time. If you have any questions, don't hesitate to ask! Happy math-ing!