Function Translation: Find G(10) Easily

Hey guys! Ever wondered how shifting a graph around affects its equation? Let's dive into a super common problem in math: translations of quadratic functions. We're going to break down a problem step-by-step, so you'll not only get the answer but also understand the why behind it. Trust me, once you get this, you’ll feel like a math whiz!

The Problem at Hand

So, we have this function: f(x) = x² + 2x. It's a classic quadratic function, which means its graph is a parabola – that U-shaped curve we all know and love (or maybe tolerate, haha). Now, this parabola is getting a makeover – it's being translated. Think of it like sliding the graph around on the coordinate plane. The translation is given by T = [8, 2]. What does this mean? It means we're moving the graph 8 units to the right and 2 units up. Our mission, should we choose to accept it, is to find the equation of the translated function, which we'll call g(x), and then figure out the value of g(10). Buckle up, let's get started!

Understanding Quadratic Functions and Transformations

Before we jump into the nitty-gritty of this specific problem, let's take a moment to zoom out and understand the bigger picture. At its heart, a quadratic function is a polynomial function with the highest power of the variable being 2. The general form looks like this: f(x) = ax² + bx + c, where a, b, and c are constants. These constants play a crucial role in determining the shape and position of the parabola. The coefficient a dictates whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and also influences its width. The b and c coefficients, in conjunction with a, determine the parabola's vertex (the minimum or maximum point) and its position on the coordinate plane.

Now, let's talk about transformations. These are operations that alter the shape or position of a graph. Translations, our focus today, are just one type of transformation. Think of them as sliding the graph without rotating or resizing it. Other common transformations include reflections (flipping the graph), stretches (making it wider or narrower), and compressions (squishing it). Each transformation has a specific effect on the function's equation. For instance, a vertical translation (moving the graph up or down) involves adding or subtracting a constant to the function's output, while a horizontal translation (moving the graph left or right) involves modifying the input variable x.

In our case, we're dealing with a translation, which means we're shifting the entire parabola. The key to handling translations is understanding how they affect the function's input. When we translate a function horizontally, we're essentially changing the x values that produce the same y values. For example, if we shift a graph 3 units to the right, the point that was originally at x = 2 will now be at x = 5. Similarly, a vertical translation shifts the y values directly.

Decoding the Translation Vector

The translation T = [8, 2] is our roadmap for shifting the graph. The first number, 8, tells us how far to move horizontally. Since it's positive, we're moving 8 units to the right. If it were negative, we'd move to the left. The second number, 2, tells us the vertical shift. Again, positive means up, so we're moving 2 units up. This vector notation is a neat way to summarize both horizontal and vertical movements in a single, compact form. Think of it as a set of instructions for where to relocate our parabola.

Now, how do we translate this geometric shift into an algebraic change in our function? This is where the magic happens. To shift a function f(x) horizontally by h units, we replace x with (x - h). So, a shift of 8 units to the right means we'll replace x with (x - 8). To shift vertically by k units, we add k to the entire function. A shift of 2 units up means we'll add 2 to f(x). Combining these two ideas is the key to finding the equation of g(x). This understanding of translation vectors and their algebraic counterparts is crucial for mastering transformations of functions. It's like learning the secret code to manipulating graphs, allowing you to predict and control their behavior with precision.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this problem! We know that f(x) = x² + 2x and T = [8, 2]. Remember, T tells us to shift 8 units to the right and 2 units up. To do this, we'll follow these steps:

1. Horizontal Shift: Replace x in f(x) with (x - 8). This gives us a new expression: (x - 8)² + 2(x - 8). This step is crucial because it directly addresses the horizontal component of the translation. By substituting (x - 8) for x, we're essentially telling the function to evaluate itself as if it were 8 units to the left, effectively shifting the graph 8 units to the right.

2. Vertical Shift: Add 2 to the entire expression from step 1. So, we have: (x - 8)² + 2(x - 8) + 2. This addition accounts for the vertical translation. Adding 2 to the entire function raises the graph by 2 units, ensuring that every point on the translated parabola is 2 units higher than its corresponding point on the original parabola.

3. Simplify: Expand and simplify the expression to get the equation for g(x). Let's do the algebra:

   • (x - 8)² = x² - 16x + 64
   • 2(x - 8) = 2x - 16
   • Putting it all together: x² - 16x + 64 + 2x - 16 + 2
   • Combining like terms: g(x) = x² - 14x + 50 Whoa! We've transformed our original function into its translated form. This simplified equation, g(x) = x² - 14x + 50, is the algebraic representation of our parabola after it has been shifted 8 units to the right and 2 units up. It encapsulates the geometric transformation in a concise mathematical expression.

4. Evaluate g(10): Now for the final piece! We need to find g(10). Simply substitute x = 10 into our equation for g(x):

   • g(10) = (10)² - 14(10) + 50
   • g(10) = 100 - 140 + 50
   • g(10) = 10 There you have it! We've successfully evaluated the translated function at x = 10, and the result is g(10) = 10. This final step demonstrates the practical application of our transformed equation. By plugging in a specific value for x, we can determine the corresponding y value on the translated parabola.

Identifying the Correct Answer

Looking back at our multiple-choice options, none of them directly match our simplified equation g(x) = x² - 14x + 50. However, the question asks for the value of g(10), which we calculated to be 10. Since none of the provided equations, when evaluated at x=10, yield 10, it seems there might be an error in the options provided. Our step-by-step solution, however, is mathematically sound, leading us to the correct function after translation and the accurate value of g(10).

It's crucial to remember that in mathematics, precision is key. While multiple-choice questions can sometimes have errors, the underlying principles and methods remain constant. By meticulously applying the rules of transformations and algebraic manipulation, we can arrive at the correct answer, even if it doesn't perfectly align with the given options. This process of problem-solving not only enhances our mathematical skills but also cultivates a critical and analytical mindset.

Key Takeaways

• Translations shift a graph without changing its shape.
• A translation vector T = [h, k] means a horizontal shift of h units and a vertical shift of k units.
• To shift f(x) horizontally by h units, replace x with (x - h).
• To shift f(x) vertically by k units, add k to the entire function.
• Always simplify your equations after transformations.

Practice Makes Perfect

The best way to master translations (or any math topic, really) is to practice! Try these variations:

1. Change the original function f(x). What if it was f(x) = 2x² - x + 3?
2. Use a different translation vector. How would the answer change if T = [-2, 5]?
3. Try combining translations with other transformations, like reflections or stretches. This is where things get really interesting!

By working through different examples, you'll build a solid understanding of how translations work and gain confidence in your ability to tackle these problems. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively.

Why This Matters

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