Curve Image After Translation And Dilation: A Math Solution

Oct 17, 2025 by ADMIN 60 views

Hey guys, ever wondered how a curve changes when you shift it and then stretch it out? Let's dive into a cool math problem where we'll figure out exactly that! We're going to take a curve, move it around, and then dilate it. It sounds complex, but trust me, we'll break it down step by step so it's super easy to understand. Our main goal is to find the new equation of the curve after these transformations. So, grab your pencils, and let's get started!

Understanding Transformations: Translation and Dilation

Before we jump into the specific problem, let’s make sure we're all on the same page about what translation and dilation actually mean in the world of math.

Translation: Shifting the Curve

In simple terms, translation is like sliding a shape or curve from one place to another without rotating or resizing it. Imagine you have a drawing on a piece of paper, and you simply move that paper across your desk. The drawing itself stays exactly the same, just its location changes. Mathematically, we represent a translation using a vector. For example, the translation $T = egin{pmatrix} 3 \ -1 hink{pmatrix}$ means we're shifting the curve 3 units to the right (positive x-direction) and 1 unit down (negative y-direction).

Think of it like this: every point on the original curve moves exactly the same distance and in the same direction. If a point (x, y) is on the original curve, after the translation, it will move to a new point (x + 3, y - 1). This is a crucial concept to grasp because it forms the basis of how we'll solve our problem. We're essentially figuring out how each point on the curve is affected by this shift.

Dilation: Stretching or Compressing the Curve

Dilation, on the other hand, changes the size of the curve. It's like zooming in or out on a picture. We specify a center of dilation (a fixed point) and a scale factor. In our problem, the dilation is [O, 3], which means the center of dilation is the origin (0, 0), and the scale factor is 3. This means the curve will be stretched away from the origin by a factor of 3. If the scale factor were between 0 and 1, the curve would be compressed instead.

Here’s the key: each point on the original curve moves along a line that connects it to the center of dilation, and its distance from the center changes by the scale factor. So, if a point (x, y) is on the original curve, after the dilation [O, 3], it will move to a new point (3x, 3y). The coordinates are simply multiplied by the scale factor. Understanding this scaling effect is super important for finding the final equation of the transformed curve.

Now that we've got a solid understanding of translation and dilation, we're well-equipped to tackle the problem at hand. We know how each transformation affects the coordinates of points on the curve, which is the key to finding the equation of the new curve.

Problem Breakdown: Finding the Image of the Curve

Okay, let's get back to the specific question. We have the curve defined by the equation $y = x^2 - 3x - 5$ , and we want to find its image after a translation and a dilation. Remember, the translation is $T = egin{pmatrix} 3 \ -1 hink{pmatrix}$ , and the dilation is [O, 3]. We're going to tackle this in two main steps: first, we'll find the equation of the curve after the translation, and then we'll apply the dilation to that result.

Step 1: Translation

As we discussed earlier, the translation $T = egin{pmatrix} 3 \ -1 hink{pmatrix}$ shifts every point (x, y) on the curve to a new point (x', y') where:

x' = x + 3
y' = y - 1

Our goal here is to find the equation of the translated curve in terms of x' and y'. To do this, we need to express the original x and y in terms of x' and y'. We can easily rearrange the equations above:

x = x' - 3
y = y' + 1

Now, we substitute these expressions for x and y into the original equation of the curve, $y = x^2 - 3x - 5$ . This is where the magic happens! We're replacing the old coordinates with the new ones to get the equation of the translated curve:

y' + 1 = (x' - 3)^2 - 3(x' - 3) - 5

Next, we need to simplify this equation. Expanding the terms and combining like terms will give us the equation of the translated curve in a cleaner form. This is a crucial step, so let's take our time and do it carefully:

y' + 1 = (x'^2 - 6x' + 9) - (3x' - 9) - 5 y' + 1 = x'^2 - 6x' + 9 - 3x' + 9 - 5 y' + 1 = x'^2 - 9x' + 13

Finally, we isolate y' to get the equation in the standard form:

y' = x'^2 - 9x' + 12

So, this is the equation of the curve after the translation. We've successfully shifted the curve, and now we know exactly what it looks like in its new position. But we're not done yet! We still need to apply the dilation.

Step 2: Dilation

Now, let's apply the dilation [O, 3] to the translated curve. Remember, this means we're stretching the curve away from the origin by a factor of 3. If a point (x', y') is on the translated curve, after the dilation, it will move to a new point (x'', y'') where:

x'' = 3x'
y'' = 3y'

Just like with the translation, we need to express x' and y' in terms of x'' and y'' so we can substitute them into the equation of the translated curve:

x' = x'' / 3
y' = y'' / 3

Now, we substitute these expressions into the equation we found in step 1, which was y' = x'^2 - 9x' + 12. Get ready for another round of substitutions and simplifications!

y'' / 3 = (x'' / 3)^2 - 9(x'' / 3) + 12

Let's simplify this equation step by step:

y'' / 3 = x''^2 / 9 - 3x'' + 12

To get rid of the fraction on the left side, we multiply both sides of the equation by 3:

y'' = x''^2 / 3 - 9x'' + 36

And there you have it! This is the equation of the curve after both the translation and the dilation. We've successfully navigated both transformations, and we've arrived at the final answer.

Final Answer: The Image of the Curve

The image of the curve $y = x^2 - 3x - 5$ after the translation $T = egin{pmatrix} 3 \ -1 hink{pmatrix}$ followed by the dilation [O, 3] is given by the equation:

$y'' = rac{1}{3}x''^2 - 9x'' + 36$

Of course, we can drop the double primes and simply write the equation as:

$y = rac{1}{3}x^2 - 9x + 36$

So, that's the final answer! We started with a curve, shifted it, stretched it, and found the new equation that describes its final position and shape. You guys did great following along! This problem demonstrates the power of transformations in mathematics and how we can use them to manipulate curves and shapes in a predictable way.

Key Takeaways and Further Practice

Let's quickly recap what we've learned and how you can practice these skills further.

Key Concepts Revisited

Translation: Shifting a curve without changing its size or shape.
Dilation: Stretching or compressing a curve with respect to a center point.
Substitution: The key technique for finding the equation of the transformed curve. We express the original coordinates in terms of the new coordinates and substitute them into the original equation.
Simplification: Crucial for getting the final equation in a clean and understandable form.

Practice Makes Perfect

To really master these concepts, try working through similar problems. Here are a few ideas:

Change the order of transformations: What happens if you dilate the curve first and then translate it? Does the final equation change? This is a great way to test your understanding.
Use different translation vectors and scale factors: Experiment with different values for the translation and dilation. This will help you see how these parameters affect the final image of the curve.
Try different curves: Work with different types of curves, such as lines, circles, or other parabolas. The process is the same, but it will give you more practice with different equations.

Transformations are a fundamental concept in geometry and are used in many areas of mathematics and computer graphics. The more you practice, the more comfortable you'll become with these techniques. Keep up the great work, guys!