Finding F(x) And Transformations: A Step-by-Step Guide

by SLV Team 55 views

Hey guys! Let's dive into some cool math problems. We're gonna break down how to find a function, f(x), when we know its shifted version, g(x). We'll also explore how functions get moved around – it's like a mathematical dance! Ready? Let's get started.

Understanding Function Transformations

Function transformations are all about how a function's graph changes position and shape on a coordinate plane. These changes can involve shifts (moving the graph up, down, left, or right), stretches/compressions (making the graph taller or wider), and reflections (flipping the graph across an axis). Understanding these transformations is key to solving our problems. We'll mainly focus on translations, which involve shifting the graph without changing its shape.

Translations: Shifts and Movements

Translations are like sliding the function's graph around. There are two main types: horizontal shifts (left or right) and vertical shifts (up or down). When we're talking about horizontal shifts, we're changing the x-values, and when we're talking about vertical shifts, we're changing the y-values. This is important to remember because it affects how we manipulate the equation of our function.

  • Horizontal Shifts:
    • To shift a graph c units to the left, we replace x with (x + c).
    • To shift a graph c units to the right, we replace x with (x - c).
  • Vertical Shifts:
    • To shift a graph c units up, we add c to the entire function, i.e., f(x) + c.
    • To shift a graph c units down, we subtract c from the entire function, i.e., f(x) - c.

The Core Concept

The fundamental idea here is that if we know how f(x) has been transformed to get g(x), we can reverse those transformations to find f(x). We will be using this concept to solve the problems below.

Problem 1: Finding f(x) After Shifts

Problem: Given that g(x) = x² + 2x - 8 is the result of shifting f(x) 4 units to the left and 3 units up, determine f(x). This problem requires us to work backward from the transformed function to find the original one. We'll use the transformations we just learned to unravel the shifts and reveal the original function.

Step-by-Step Solution

  1. Understand the Shifts:
    • Left 4 units: This means the x-values in f(x) were originally (x + 4).
    • Up 3 units: This means we added 3 to the entire function.
  2. Reverse the Vertical Shift: Since g(x) is shifted up by 3, we subtract 3 from g(x) to reverse this shift. So, we get x² + 2x - 8 - 3 = x² + 2x - 11.
  3. Reverse the Horizontal Shift: Since the graph was shifted 4 units to the left, it means that x has been replaced with (x + 4). To reverse this, we replace x with (x - 4) in the expression we got in the previous step.
    • So, replace x with (x - 4) in x² + 2x - 11:
      • f(x) = (x - 4)² + 2(x - 4) - 11
  4. Simplify and Find f(x):
    • Expand and simplify the expression:
      • f(x) = (x² - 8x + 16) + (2x - 8) - 11
      • f(x) = x² - 8x + 16 + 2x - 8 - 11
      • f(x) = x² - 6x - 3
    • Therefore, f(x) = x² - 6x - 3

So, by carefully reversing the vertical and horizontal shifts, we found the original function f(x). Nice job, guys!

Problem 2: Determining the Shift of f(x)

Problem: Given f(x) = x², determine how f(x) is shifted if the result of the shift is g(x) = x² - 6x + 5. This problem is the reverse of the first one; we start with the original and the transformed function and need to figure out the shifts that took place.

Step-by-Step Solution

  1. Rewrite g(x) to Reveal Transformations:
    • We can rewrite g(x) by completing the square to identify the shifts more easily.
    • g(x) = x² - 6x + 5
    • To complete the square, take half the coefficient of x (-6), square it (9), and add and subtract it:
      • g(x) = (x² - 6x + 9) - 9 + 5
      • g(x) = (x - 3)² - 4
  2. Analyze the Transformations:
    • Horizontal Shift: The (x - 3) inside the parentheses indicates a shift 3 units to the right.
    • Vertical Shift: The - 4 at the end indicates a shift 4 units down.
  3. Conclusion: The function f(x) = x² is shifted 3 units to the right and 4 units down to obtain g(x) = x² - 6x + 5.

Summarizing the Problem

In this problem, we analyzed the change from f(x) to g(x) by rewriting g(x) in a form that clearly showed the transformations. This approach of completing the square is super useful for identifying horizontal and vertical shifts.

Key Takeaways and Tips

Key Takeaways

  • Understand the Direction: Left and right shifts involve changing the x-values, and up and down shifts involve changing the entire function (adding or subtracting).
  • Reverse the Process: When finding f(x) from g(x), reverse the shifts in the opposite direction.
  • Completing the Square: This is an invaluable tool for rewriting quadratic functions to reveal their transformations.

Tips for Success

  • Practice: Work through various examples to get comfortable with the concepts.
  • Visualize: Sketching the graphs can help you understand the shifts.
  • Be Careful with Signs: Pay close attention to the signs (plus or minus) when determining the direction of the shifts.

Final Thoughts

So, there you have it, guys! We have explored how function transformations work and how to find f(x) when given g(x) and vice-versa. Remember, the core of these problems is understanding the relationship between the original function and its shifted form. Keep practicing, and you'll become a pro at these problems in no time. If you have any questions, feel free to ask. Keep up the great work, and happy calculating!