Which translation maps the vertex of the graph of the function f (x) = x2 onto the vertex of the function g (x) = x2 - 10x + 2?

The translation that maps the vertex of the graph of the function f (x) = x² onto the vertex of the function g (x) = x² - 10x + 2 is 5 units to the right and 23 units down.

Explanation:

1) Vertex form of the function that represents a parabola.

The general form of a quadratic equation is Ax² + Bx + C = 0, where A ≠ 0, and B and C may be any real number. And the graph of such equation is a parabola with a minimum or maximum value at its vertex.

The vertex form of the graph of such function is: A (x - h) ² + k

Where, A a a stretching factor (in the case |A| > 1) or compression factor (in the case |A| < 1) factor.

2) Find the vertex of the first function, f (x) = x²

This is the parent function, for which, by simple inspection, you can tell h = 0 and k = 0, i. e. the vertex of f (x) = x² is (0,0).

3) Find teh vertex of the second function, g (x) = x² - 10x + 2

The method is transforming the form of the function by completing squares:

Subtract 2 from both sides: g (x) - 2 = x² - 10x

Add the square of half of the coefficient of x (5² = 25) to both sides: g (x) - 2 + 25 = x² - 10x + 25

Simplify the left side and factor the right side: g (x) + 23 = (x - 5) ²

Subtract 23 from both sides: g (x) = (x - 5) ² - 23

That is the searched vertex form: g (x) = (x - 5) ² - 23.

From that, using the rules of translation you can conclude immediately that the function f (x) was translated 5 units horizontally to the right and 23 units vertically downward.

Also, by comparison with the verex form A (x - h) ² + k, you can conclude that the vertex of g (x) is (5, - 23), and that means that the vertex (0,0) was translated 5 units to the right and 23 units downward.