Transformations Of F(x) = X^2 To G(x): Explained!

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Transformations of f(x) = x^2 to g(x) = -5x^2 + 100x - 450: A Comprehensive Guide

Hey guys! Today, we're diving deep into the fascinating world of graph transformations. Specifically, we're going to break down how the graph of the simple quadratic function f(x) = x² can be transformed into the more complex graph of g(x) = -5x² + 100x - 450. This involves understanding shifts, stretches, and reflections, so buckle up and let's get started!

Understanding the Parent Function: f(x) = x²

Before we jump into the transformations, it's crucial to have a solid grasp of the parent function, f(x) = x². This is the basic parabola, a U-shaped curve that opens upwards. Its vertex, the lowest point on the graph, is located at the origin (0, 0). Knowing this foundation allows us to easily identify how transformations alter this basic shape and position. The key characteristics of f(x) = x² that we'll be looking for changes in are the vertex position, the direction the parabola opens, and how wide or narrow it is. Understanding these fundamental aspects of the parent function is essential for accurately identifying and interpreting transformations applied to it.

Remember, the graph of f(x) = x² is symmetrical about the y-axis. This symmetry is also something that might be affected by transformations, particularly horizontal shifts. So, as we analyze the transformed function g(x), keep in mind the original symmetry and how it might have changed. Visualizing the parent function and its key features in your mind will make it much easier to trace the steps of transformation that lead to the final graph of g(x). This mental image serves as a reference point, allowing you to pinpoint the exact alterations made to the original function. Mastering this concept is crucial for excelling in understanding function transformations.

Analyzing the Transformed Function: g(x) = -5x² + 100x - 450

Now, let's tackle the transformed function, g(x) = -5x² + 100x - 450. The first thing we need to do is rewrite this equation in vertex form. Vertex form is super helpful because it directly reveals the vertex of the parabola and any vertical stretches or compressions. The general form of vertex form is g(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola and 'a' determines the vertical stretch/compression and reflection. To get our function into this form, we'll use a technique called completing the square.

Completing the square involves manipulating the quadratic expression to create a perfect square trinomial. First, factor out the coefficient of the x² term (which is -5 in our case) from the first two terms: g(x) = -5(x² - 20x) - 450. Next, we need to add and subtract a value inside the parentheses that will complete the square. This value is (b/2)², where 'b' is the coefficient of the x term inside the parentheses (which is -20). So, (-20/2)² = 100. We add and subtract this value inside the parentheses: g(x) = -5(x² - 20x + 100 - 100) - 450. Now, the first three terms inside the parentheses form a perfect square trinomial: g(x) = -5((x - 10)² - 100) - 450. Distribute the -5: g(x) = -5(x - 10)² + 500 - 450. Finally, simplify to get the vertex form: g(x) = -5(x - 10)² + 50. This is a key step, as it unlocks all the transformation information we need!

Identifying the Transformations

With g(x) in vertex form, g(x) = -5(x - 10)² + 50, we can now easily identify the transformations applied to f(x) = x². Let's break it down:

  1. Vertical Stretch/Compression and Reflection: The '-5' in front of the squared term indicates two things: a vertical stretch by a factor of 5 (the absolute value of -5) and a reflection across the x-axis (the negative sign). This means the parabola opens downwards instead of upwards, and it's significantly narrower than the parent function. The vertical stretch makes the graph taller and skinnier, while the reflection flips it upside down.

  2. Horizontal Shift: The '(x - 10)' term indicates a horizontal shift. Specifically, it's a shift to the right by 10 units. Remember, it's the opposite of what you might intuitively think! The graph is shifted 10 units to the right because we're replacing 'x' with '(x - 10)'. This moves the vertex of the parabola 10 units along the x-axis.

  3. Vertical Shift: The '+ 50' at the end of the equation indicates a vertical shift upwards by 50 units. This moves the entire graph, including the vertex, 50 units up the y-axis. So, the vertex has been shifted both horizontally and vertically.

So, to recap, the transformations are: a vertical stretch by a factor of 5, a reflection across the x-axis, a horizontal shift of 10 units to the right, and a vertical shift of 50 units upwards. Understanding the order in which these transformations are applied is crucial. The stretches/compressions and reflections are usually considered before the shifts.

Connecting the Transformations to the Options

Now that we've identified the transformations, let's relate them to the multiple-choice options provided (which you didn't explicitly include, but we can still address in general terms):

  • Option A (Shifted up 50 units): This is a correct transformation, as we identified a vertical shift upwards by 50 units.
  • Option B (Shifted left 10 units): This is incorrect. We identified a horizontal shift to the right by 10 units, not to the left.
  • A Correct Option (Horizontal Shift Right 10 Units): This should be one of the correct answers as we discussed above. The (x-10) term in the vertex form directly indicates a shift of 10 units to the right.
  • A Correct Option (Vertical Stretch and Reflection): Another correct answer would describe the vertical stretch by a factor of 5 and the reflection over the x-axis due to the -5 coefficient. Options describing these transformations should be selected.

Therefore, to accurately answer the original multiple-choice question, you would need to select the options that correctly describe the vertical shift upwards by 50 units, the horizontal shift to the right by 10 units, the vertical stretch by a factor of 5, and the reflection across the x-axis. It's important to read each option carefully and match it to the transformations we identified from the vertex form of the equation. Being meticulous in this matching process is key to success.

Visualizing the Transformations

One of the best ways to solidify your understanding of these transformations is to visualize them. Imagine starting with the graph of f(x) = x². First, reflect it across the x-axis so it opens downwards. Then, stretch it vertically, making it narrower. Next, shift the entire graph 10 units to the right. Finally, shift it 50 units upwards. The resulting graph should match the graph of g(x) = -5x² + 100x - 450. Using graphing software or even sketching the transformations step-by-step can be incredibly helpful. Visualization turns abstract concepts into concrete mental images, boosting comprehension.

Tips for Mastering Graph Transformations

Here are a few extra tips to help you master graph transformations:

  • Memorize the Parent Functions: Knowing the basic shapes of common functions like x², x³, √x, |x|, and 1/x is essential. They are the building blocks for more complex graphs.
  • Practice, Practice, Practice: The more you work through transformation problems, the more comfortable you'll become with identifying them. Work through various examples with different types of functions and transformations.
  • Use Graphing Tools: Graphing calculators or online tools can be invaluable for visualizing transformations and checking your answers. Use these tools to explore how changes in the equation affect the graph.
  • Understand the Order of Operations: As mentioned earlier, the order in which transformations are applied matters. Generally, stretches/compressions and reflections are applied before shifts.
  • Relate to Real-World Examples: Transformations have real-world applications in areas like physics, engineering, and computer graphics. Thinking about these applications can make the concepts more meaningful.

Conclusion

Understanding graph transformations is a fundamental concept in mathematics, and it's crucial for success in algebra and beyond. By mastering the techniques we've discussed, such as converting to vertex form, identifying the individual transformations, and visualizing the process, you'll be well-equipped to tackle any transformation problem that comes your way. Remember, practice makes perfect, so keep working at it, and you'll become a transformation pro in no time! You got this!