The graph of f(x) = x2 is translated to form g(x) = (x – 5)2 + 1. On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4). Which graph represents g(x)? On a coordinate plane, a parabola opens up. It goes through (2, 10), has a vertex at (5, 1), and goes through (8, 10). On a coordinate plane, a parabola opens up. It goes through (2, 8), has a vertex at (5, negative 11), and goes through (8, 8). On a coordinate plane, a parabola opens up. It goes through (negative 8, 10), has a vertex at (negative 5, 1), and goes through (negative 2, 10). On a coordinate plane, a parabola opens up. It goes through (negative 8, 8), has a vertex at (negative 5, negative 11), and goes through (negative 2, 8).

The transformation of a function may involve any change. When the graph of f(x) is transformed into 5 units to the right and one unit up, then the function g(x) is obtained.

How does the transformation of a function happen?

The transformation of a function may involve any change.

Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs) etc.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Horizontal shift (also called phase shift):

Left shift by c units, y=f(x+c) (same output, but c units earlier)
Right shift by c units, y=f(x-c)(same output, but c units late)

Vertical shift

Up by d units: y = f(x) + d
Down by d units: y = f(x) - d

Stretching:

Vertical stretch by a factor k: y = k \times f(x)
Horizontal stretch by a factor k: y = f(x/k)

Given the function f(x)=x², which is transformed to g(x)=(x-5)²+1, therefore, the graph of both the functions are given below.

When the graph of f(x) is transformed into 5 units to the right and one unit up, then the function g(x) is obtained.

Learn more about Transforming functions:

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