Mastering Function Transformations: Translation, Scaling, and Reflection

3 different ways to transform a function & how they affect graph

There are several ways to transform a function, each of which can have a different effect on the graph

There are several ways to transform a function, each of which can have a different effect on the graph. Here, I will explain three common transformations: translation, scaling, and reflection.

1. Translation: This transformation involves moving the graph of a function horizontally or vertically. To translate a function horizontally, you can add or subtract a value to the input variable (x). For example, if you have the function f(x), and you want to shift it 2 units to the right, you would use the function g(x) = f(x – 2). Similarly, to translate a function vertically, you add or subtract a value to the output variable (y). For instance, if you have the function f(x) and want to shift it 3 units up, you would use the function g(x) = f(x) + 3.

2. Scaling: Scaling involves stretching or compressing a function horizontally or vertically. To scale a function horizontally, you can multiply or divide the input variable (x) by a value. If you have the function f(x) and you want to stretch it horizontally by a factor of 2, you would use the function g(x) = f(2x). On the other hand, if you want to compress it horizontally by 3, you would use the function g(x) = f(x/3). Scaling vertically is done by multiplying or dividing the output variable (y). For example, if you want to stretch the function f(x) vertically by a factor of 3, you would use the function g(x) = 3f(x). If you want to compress it vertically by half, you would use the function g(x) = (1/2)f(x).

3. Reflection: Reflection involves flipping the graph of a function over a line. There are two main lines used for reflection: the x-axis and the y-axis. To reflect a function over the x-axis, you negate the output variable (y). For example, if you have the function f(x), and you want to reflect it over the x-axis, you would use the function g(x) = -f(x). This will cause the graph to appear upside down. Similarly, to reflect a function over the y-axis, you negate the input variable (x). If you have the function f(x), and you want to reflect it over the y-axis, you would use the function g(x) = f(-x). This will cause the graph to appear as a mirror image on the other side of the y-axis.

Remember, these transformations can be combined and applied to any type of function. Each transformation has a specific effect on the graph, altering its shape, position, and orientation. By understanding these transformations, you can manipulate functions to create new ones and explore various mathematical concepts.

More Answers: