which equation represents the transformed function below

Guri Bee

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22-10-2024

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Unraveling Transformations: Finding the Equation of a Transformed Function

Transforming functions is a fundamental concept in mathematics, allowing us to manipulate and analyze graphs with ease. But how do we find the equation of a transformed function given its graph? Let's delve into this question using a real-world example and some helpful insights from GitHub.

Understanding the Basics

Imagine you have a basic function, let's say *f(x) = x². * This function represents a simple parabola opening upwards. Now, let's say we want to move this parabola 3 units to the right and 2 units down. How would the equation change?

This is where transformations come into play. We can apply specific changes to the original function to achieve the desired transformation. In our example, we need to:

1. Horizontal Shift: To shift the graph 3 units to the right, we subtract 3 from the input x, resulting in f(x - 3).
2. Vertical Shift: To shift the graph 2 units down, we subtract 2 from the entire function, resulting in f(x - 3) - 2.

Therefore, the equation of the transformed function becomes: (x - 3)² - 2.

Finding the Equation from a Graph

Now, let's analyze a specific example, where the graph of a transformed function is given, and we need to find the corresponding equation.

Example:

Problem: A graph shows a function that looks like a parabola opening upwards, shifted 2 units to the left and 1 unit up from the original parabola y = x². Find the equation of the transformed function.

Solution:

Based on our understanding of transformations, we can deduce the following:

1. Horizontal Shift: The graph is shifted 2 units to the left, so we add 2 to the input x, resulting in f(x + 2).

2. Vertical Shift: The graph is shifted 1 unit up, so we add 1 to the entire function, resulting in f(x + 2) + 1.

Therefore, the equation of the transformed function is: (x + 2)² + 1.

Additional Tips from GitHub:

• GitHub User "math_wizard": "Remember that the order of transformations can be important. Apply the horizontal shift first, followed by the vertical shift."
• GitHub User "graph_geek": "If the graph is reflected about the x-axis or y-axis, multiply the function by -1 or add a negative sign in front of the input x, respectively."

Practical Examples:

• Motion: The position of an object moving at constant acceleration can be modeled using a parabolic function. By understanding transformations, we can predict the object's position at different times.

• Data Analysis: Many statistical models use transformed functions to analyze data and understand relationships between variables.

Conclusion:

Finding the equation of a transformed function is a valuable skill in mathematics and other fields. By understanding the basics of transformations, we can easily analyze and manipulate graphs to solve problems and explore complex relationships. With the help of resources like GitHub and practical examples, we can confidently navigate the world of transformed functions.