Translating the Quadratic Function: Finding g(x) if f(x) = x^2

Quadratic functions are an important part of mathematics and are used in a variety of fields. The quadratic function is a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

In this blog post, we’ll be focusing on translating the quadratic function and finding g(x) if f(x) = x^2.

First off, let’s define what we mean by “translating” a function. Translating a function means changing its position on the coordinate plane without changing its shape. This is done by adding or subtracting a constant to the function.

Now, let’s apply this concept to the quadratic function f(x) = x^2. To find g(x), we need to translate f(x) by a constant, say h, to get g(x) = (x-h)^2.

To see how this works, let’s take an example. Suppose we want to find g(x) if f(x) = x^2 and h = 3. To translate f(x), we subtract 3 from x to get g(x) = (x-3)^2.

Now, let’s plot the graphs of f(x) and g(x) on the same coordinate plane. We’ll use Desmos, a free online graphing calculator.

As you can see from the graph, g(x) is simply f(x) shifted 3 units to the right. This is because when we substitute x-3 for x in f(x), we get (x-3)^2 = x^2 – 6x + 9. This means that g(x) has the same shape as f(x), but it’s shifted 3 units to the right and has a vertex at (3,0).

In conclusion, translating the quadratic function is a useful tool in mathematics. To find g(x) if f(x) = x^2, we simply need to subtract a constant from x and square the result. This gives us a new function with the same shape as the original, but shifted on the coordinate plane.