Transformations Of X^5: Find The New Function

by Alex Johnson 46 views

Hey math enthusiasts! Ever wondered how a function like f(x) = x^5 changes when you twist and turn it with transformations? Let's dive deep into understanding how vertical stretches, upward shifts, and leftward translations affect this powerful fifth-degree function. We'll break down each step so that by the end, you'll be able to confidently determine the new function resulting from a sequence of transformations.

Understanding the Basics: Parent Function and Transformations

Our starting point is the parent function f(x) = x^5. This is the foundation upon which we'll build. Think of it as the original, untouched version. The exponent of 5 means that the function grows very rapidly as x moves away from zero, and it has a unique S-shape. Now, let's look at the transformations involved in this problem: a vertical stretch, an upward translation, and a leftward translation.

  • Vertical Stretch: This transformation changes the steepness of the function. Multiplying the function by a constant (in this case, 3) stretches the graph vertically, making it steeper.
  • Upward Translation: This shift moves the entire graph upwards along the y-axis. Adding a constant to the function achieves this.
  • Leftward Translation: This shift moves the entire graph to the left along the x-axis. This one involves changing the x inside the function. Adding a constant inside the function (i.e., replacing x with (x + constant)) shifts the graph to the left.

It’s like molding a piece of clay. You can stretch it, lift it, and move it side to side. Each action changes the shape and position of the clay. Similarly, each transformation alters the parent function, creating a new, transformed function.

Step-by-Step Transformation of f(x) = x^5

Let’s apply the transformations step-by-step, to see how the function evolves. This method is the key to understanding how the transformations stack up. It helps you keep track of each change and how it affects the final result.

Step 1: Vertical Stretch

The first transformation is a vertical stretch by a factor of 3. This means we multiply the entire function by 3. Our new function after this step is 3 * f(x) = 3x^5. This transformation makes the graph steeper. If you were to plot both the original and this transformed function, you would see the change. The new function grows more rapidly compared to the parent.

Step 2: Upward Translation

Next, we translate the graph up by 1 unit. This means we add 1 to the function. So, our function becomes 3x^5 + 1. This step simply shifts the entire graph upwards, so every point on the graph is now 1 unit higher than before.

Step 3: Leftward Translation

The final transformation is a leftward translation by 2 units. Here’s where it gets a bit trickier. To move the graph left, we replace x with (x + 2) inside the function. So we take the function from the previous step which is 3x^5 + 1 and substitute the x so we get 3(x + 2)^5 + 1. This means the entire graph is shifted 2 units to the left. The + 2 inside the parentheses has the opposite effect of what you might expect; it moves the graph in the negative direction, i.e., to the left.

So, after all the transformations, the final function is 3(x + 2)^5 + 1. This is the result of the vertical stretch, the upward shift, and the leftward translation applied in sequence.

Matching the Result with the Options

Now, let's match our derived function with the answer choices. Our final function is 3(x + 2)^5 + 1. Comparing this with the options provided:

  • A. f(x) = (3x - 2)^5 + 1: This is incorrect because it incorrectly applies the transformations.
  • B. f(x) = 3(x + 2)^5 + 1: This is correct! It accurately reflects the transformations.
  • C. f(x) = 3(x - 1)^5 + 2: This option isn't equivalent to our result, as it doesn't represent the applied transformations correctly.
  • D. f(x) = 3(x - 2)^5 + 1: This option is incorrect because it gets the sign wrong in the horizontal translation.

Therefore, the correct answer is option B.

Why Understanding Transformations Matters

Understanding transformations is a fundamental skill in mathematics, not just for functions like f(x) = x^5, but for all kinds of functions. It allows you to visualize and predict how changes to an equation will affect its graph. This is incredibly useful in various fields, including:

  • Calculus: Where you need to understand the behavior of functions to analyze rates of change and areas.
  • Physics: Where you can model motion and other physical phenomena.
  • Engineering: Where you can design systems that respond to inputs in predictable ways.
  • Computer Graphics: Where you can manipulate images and objects.

Mastering transformations equips you with a powerful toolset for problem-solving in mathematics and beyond. It’s like having a set of tools that lets you reshape and analyze mathematical objects with ease. This skill goes beyond simply solving equations; it helps to develop a deeper understanding of mathematical relationships.

Conclusion: Transformation Triumph!

We've successfully transformed the function f(x) = x^5 through a series of steps and found the resulting function. Remember, the order of transformations is crucial. With this step-by-step approach, you can confidently tackle similar problems in the future. Keep practicing, and you will become proficient at predicting and applying transformations. The ability to manipulate and understand functions is a key element of mathematical literacy and is a skill that will be useful across a wide range of academic and professional areas. Keep exploring and happy transforming!

For more in-depth explanations and practice problems, you can visit the Khan Academy website. This is a great resource for further learning. Check it out and keep exploring the amazing world of mathematics!