Find G(x): Quadratic Function Translations Explained

by Luna Greco 53 views

Hey guys! Today, we're diving into a super interesting problem involving quadratic functions and their transformations. Specifically, we're going to figure out how to find the equation of a translated quadratic function. Imagine you have a basic parabola, f(x) = x^2, and you shift it around the coordinate plane – that's essentially what we're dealing with. Let's break it down step by step so you can ace these types of problems.

Problem Overview

Our core challenge lies in determining the equation for a new function, g(x), which is essentially a translated version of the fundamental quadratic function f(x) = x^2. Think of it like this: we're taking the basic parabola and sliding it around. The key piece of information we have is the vertex of this new parabola, g(x). We know that the vertex of g(x) is located 5 units above and 7 units to the right of the vertex of f(x). This is crucial because the vertex gives us the most important clues about the translation that has occurred. We need to translate this geometric shift into an algebraic equation, effectively capturing the movement of the parabola in mathematical terms. This involves understanding how horizontal and vertical shifts affect the equation of a quadratic function, and then applying this knowledge to the specific shifts described in the problem. By the end of this explanation, you'll not only understand how to solve this particular problem, but also the general principles behind translating quadratic functions, making you well-equipped to tackle similar challenges in the future. This foundational understanding is essential for more advanced topics in algebra and calculus, where transformations of functions play a significant role.

Understanding the Parent Function: f(x) = x^2

Let's start with the basics. The function f(x) = x^2 is our parent function, the foundation upon which we'll build our understanding of transformations. This is the most basic quadratic function, forming a symmetrical U-shaped curve called a parabola. The most important point on this parabola is its vertex, which is the point where the parabola changes direction. For f(x) = x^2, the vertex is located smack-dab at the origin, the point (0, 0). This is because when x is 0, x squared is also 0, and for any other value of x (positive or negative), x squared will be a positive number, placing the rest of the parabola above the x-axis. The parabola opens upwards, extending infinitely in both directions. Understanding the symmetry of the parabola is also key. It's symmetrical around a vertical line that passes through the vertex, known as the axis of symmetry. For f(x) = x^2, the axis of symmetry is simply the y-axis (the line x = 0). Knowing these fundamental characteristics of f(x) = x^2 – its shape, vertex, and symmetry – provides a crucial reference point when we start considering translations. When we talk about shifting or moving the parabola, we're essentially comparing its new position to this original, unshifted form. So, keep this image of f(x) = x^2 in your mind as we move forward; it's our anchor in the world of quadratic transformations.

Decoding Translations: Horizontal and Vertical Shifts

Now, let's talk about how we can move this parabola around. This is where translations come into play. Translations are basically sliding the graph of a function without changing its shape or orientation. There are two main types of translations we need to consider: horizontal shifts and vertical shifts. Think of horizontal shifts as moving the graph left or right along the x-axis, and vertical shifts as moving it up or down along the y-axis. These shifts are achieved by making adjustments inside and outside the function, respectively.

Horizontal Shifts

Horizontal shifts affect the x-coordinate. If we want to shift the graph to the right, we subtract a constant from x inside the function. Conversely, to shift the graph to the left, we add a constant to x inside the function. It might seem counterintuitive, but subtracting moves it right, and adding moves it left! For example, if we have f(x) = x^2 and we want to shift it 3 units to the right, we would replace x with (x - 3), resulting in a new function f(x - 3) = (x - 3)^2. The graph of this new function will look exactly like the original parabola, but its vertex will now be at the point (3, 0) instead of (0, 0).

Vertical Shifts

Vertical shifts are more straightforward – they affect the y-coordinate. To shift the graph up, we add a constant outside the function. To shift the graph down, we subtract a constant outside the function. For instance, if we want to shift f(x) = x^2 up by 2 units, we simply add 2 to the function, giving us f(x) + 2 = x^2 + 2. This new parabola will look identical to the original, but its vertex will be at the point (0, 2) instead of (0, 0).

Understanding these two types of shifts is crucial for manipulating functions and visualizing their graphs. They are fundamental transformations that appear in various mathematical contexts, so mastering them will definitely pay off in the long run.

The Vertex Form of a Quadratic Equation: Unlocking the Secrets

The vertex form of a quadratic equation is your best friend when dealing with translations. It's a special way of writing a quadratic equation that makes the vertex readily apparent. The general form of the vertex form is:

g(x) = a(x - h)^2 + k

Where:

  • a determines the direction the parabola opens (upwards if a is positive, downwards if a is negative) and its vertical stretch or compression.
  • (h, k) represents the coordinates of the vertex of the parabola. This is the magic! The values of h and k directly tell us how the parabola has been shifted horizontally and vertically, respectively, from the parent function f(x) = x^2.

Think of h as the horizontal shift. If h is positive, the parabola has been shifted h units to the right. If h is negative, the parabola has been shifted |h| units to the left. Similarly, k represents the vertical shift. If k is positive, the parabola has been shifted k units up. If k is negative, the parabola has been shifted |k| units down.

So, by simply looking at the vertex form of a quadratic equation, we can immediately identify the vertex and, consequently, understand how the parabola has been translated from its parent function. This is why the vertex form is such a powerful tool for analyzing and manipulating quadratic functions. It transforms the problem of finding the vertex from a potentially complex calculation into a straightforward observation. Mastering the vertex form is key to effortlessly solving translation problems and gaining a deeper understanding of quadratic functions.

Applying the Translations to Find g(x)

Okay, let's get back to our original problem. We know that g(x) is a translation of f(x) = x^2, and the vertex of g(x) is 5 units above and 7 units to the right of the vertex of f(x). Remember, the vertex of f(x) is at (0, 0).

This means the vertex of g(x) is at (7, 5). We've essentially decoded the translations: a horizontal shift of 7 units to the right and a vertical shift of 5 units up. Now, let's use the vertex form to build the equation for g(x).

We know the vertex form is g(x) = a(x - h)^2 + k, and we've determined that (h, k) = (7, 5). So, we can plug these values into the equation:

g(x) = a(x - 7)^2 + 5

Now, we need to figure out the value of a. But here's the thing: the problem tells us that g(x) is a translation of f(x). This implies that the shape of the parabola hasn't changed, only its position. The shape of the parabola is determined by the value of a. For f(x) = x^2, a is equal to 1. Since g(x) is just a translation, it maintains the same shape, so a for g(x) is also 1.

Therefore, the equation for g(x) is:

g(x) = 1(x - 7)^2 + 5

Which simplifies to:

g(x) = (x - 7)^2 + 5

And that's it! We've successfully found the equation of g(x) by understanding the translations and applying the vertex form.

Expanding and Simplifying (Optional)

While g(x) = (x - 7)^2 + 5 is a perfectly valid answer, sometimes you might need to express it in the standard quadratic form, which is ax^2 + bx + c. To do this, we simply expand and simplify the equation:

g(x) = (x - 7)(x - 7) + 5 g(x) = x^2 - 7x - 7x + 49 + 5 g(x) = x^2 - 14x + 54

So, g(x) = x^2 - 14x + 54 is another way to represent the same function. Both forms are correct, but the vertex form g(x) = (x - 7)^2 + 5 is often more useful for quickly identifying the vertex and understanding the transformations.

Key Takeaways: Mastering Quadratic Translations

Let's recap the key takeaways from this problem. Understanding these concepts will help you tackle similar problems with confidence.

  1. The Parent Function: Always start with the parent function, f(x) = x^2, as your reference point.
  2. Horizontal Shifts: Shifts inside the function (e.g., x - h) move the graph left or right (subtracting moves it right!).
  3. Vertical Shifts: Shifts outside the function (e.g., + k) move the graph up or down (adding moves it up!).
  4. The Vertex Form: g(x) = a(x - h)^2 + k is your best friend! (h, k) is the vertex, and a determines the shape and direction of the parabola.
  5. Translations Preserve Shape: If a function is simply translated, the value of a remains the same as the parent function.

By mastering these concepts, you'll be able to confidently analyze and manipulate quadratic functions, no matter how they're translated. Keep practicing, and you'll become a pro at spotting those vertices and writing those equations!

Practice Problems: Sharpen Your Skills

To solidify your understanding of quadratic translations, let's try a couple of practice problems. These will help you apply the concepts we've discussed and build your confidence.

Problem 1:

The graph of h(x) is a translation of f(x) = x^2. The vertex of h(x) is located 3 units below and 2 units to the left of the vertex of f(x). Which equation represents h(x)?

Problem 2:

A parabola has a vertex at (4, -1) and is a translation of f(x) = x^2. Write the equation of the parabola in vertex form.

Try solving these problems on your own, using the steps and concepts we've covered. Remember to identify the horizontal and vertical shifts, and then use the vertex form to construct the equation. If you get stuck, revisit the explanations above, and don't be afraid to break the problem down into smaller steps. Practice makes perfect, and the more you work with these types of problems, the more comfortable you'll become with quadratic translations. You can also explore online resources or textbooks for additional practice problems and examples. The key is to actively engage with the material and challenge yourself to apply what you've learned. Good luck, and happy problem-solving!