Question

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?

Answer 1

Answer:

• 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.