Transformations Of F(x) = X^2 To G(x) = 2x^2 - 28x + 3

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Transformations of f(x) = x^2 to g(x) = 2x^2 - 28x + 3

Let's dive into how we can transform the graph of the basic quadratic function, f(x) = x², into the more complex function g(x) = 2x² - 28x + 3. This involves understanding the different types of transformations we can apply, such as shifts (horizontal and vertical) and stretches/compressions. By the end of this explanation, you'll not only know the answer but also understand the why behind it. So, buckle up, math enthusiasts, let's get started!

Understanding Transformations

Before we tackle the specific problem, let's quickly review the common transformations that can be applied to a function's graph:

  • Vertical Shifts: Adding a constant to the function, f(x) + c, shifts the graph vertically. If c is positive, the graph shifts upward; if c is negative, it shifts downward.
  • Horizontal Shifts: Replacing x with (x - c) in the function, f(x - c), shifts the graph horizontally. If c is positive, the graph shifts to the right; if c is negative, it shifts to the left.
  • Vertical Stretches/Compressions: Multiplying the function by a constant, a f(x), stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. If a is negative, it also reflects the graph across the x-axis.
  • Horizontal Stretches/Compressions: Replacing x with (ax) in the function, f(ax), compresses the graph horizontally if |a| > 1 and stretches it if 0 < |a| < 1. If a is negative, it also reflects the graph across the y-axis.

Analyzing the Given Functions

We're given two functions:

  • f(x) = x² (the parent function)
  • g(x) = 2x² - 28x + 3

Our goal is to determine the transformations that, when applied to f(x), will result in g(x). The key here is to rewrite g(x) in a form that clearly shows the transformations. This often involves completing the square.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form a(x - h)² + k, which makes it easy to identify the vertex of the parabola and the transformations applied. Let's apply this to g(x):

  1. Factor out the coefficient of the x² term (which is 2) from the first two terms: g(x) = 2(x² - 14x) + 3
  2. Complete the square inside the parentheses. To do this, take half of the coefficient of the x term (-14), square it ((-7)² = 49), and add and subtract it inside the parentheses: g(x) = 2(x² - 14x + 49 - 49) + 3
  3. Rewrite the first three terms inside the parentheses as a squared binomial: g(x) = 2((x - 7)² - 49) + 3
  4. Distribute the 2 and simplify: g(x) = 2(x - 7)² - 98 + 3 g(x) = 2(x - 7)² - 95

Now we have g(x) in the form a(x - h)² + k. This form tells us a lot about the transformations.

Identifying the Transformations

By looking at the completed square form of g(x) = 2(x - 7)² - 95, we can identify the following transformations applied to f(x) = x²:

  1. Vertical Stretch: The factor of 2 in front of the squared term indicates a vertical stretch by a factor of 2. This makes the parabola narrower compared to f(x).
  2. Horizontal Shift: The (x - 7) term indicates a horizontal shift. Since we're subtracting 7 from x, the graph shifts 7 units to the right.
  3. Vertical Shift: The - 95 term indicates a vertical shift. Since we're subtracting 95, the graph shifts 95 units downward.

Matching with the Answer Choices

Now, let's look at the answer choices provided in the original question:

A. Shifted up 3 units B. Shifted left 7 units C. Shifted right 7 units D. Shifted down 3 units

Comparing our identified transformations with the answer choices, we see that C. shifted right 7 units matches one of the transformations we found.

Therefore, the correct answer is C.

Why Other Options are Incorrect

Let's briefly discuss why the other options are incorrect:

  • A. Shifted up 3 units: Our analysis showed a vertical shift of 95 units downward, not 3 units upward.
  • B. Shifted left 7 units: Our analysis showed a horizontal shift of 7 units to the right, not the left. Remember, (x - 7) shifts the graph to the right.
  • D. Shifted down 3 units: Again, the vertical shift was 95 units downward, not 3 units.

Key Takeaways

  • Completing the square is a powerful technique for rewriting quadratic functions in a form that reveals transformations.
  • Understanding the meaning of a, h, and k in the vertex form a(x - h)² + k is crucial for identifying transformations.
  • Horizontal shifts are counterintuitive: (x - c) shifts the graph to the right, and (x + c) shifts it to the left.
  • Always carefully compare your identified transformations with the answer choices to select the correct one.

Practice Makes Perfect

To solidify your understanding of transformations, try working through more examples. You can find practice problems online or in textbooks. The more you practice, the more comfortable you'll become with identifying transformations and manipulating functions.

Conclusion

Transforming functions can seem daunting at first, but with a solid understanding of the basic transformations and techniques like completing the square, you can confidently tackle these types of problems. Remember, it's all about breaking down the problem into smaller steps and understanding the underlying principles. Keep practicing, and you'll become a transformation master in no time! You've got this, guys! Let's keep learning and exploring the fascinating world of mathematics. Remember to always question, explore, and never stop learning! Math is not just about numbers and equations; it's about understanding patterns, relationships, and the beauty of logical thinking. So, embrace the challenge, enjoy the journey, and let's conquer math together!