Answer:
- <em>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 </em><u>5 units to the right and 23 units down.</u>
Explanation:
<u>1) </u><u>Vertex form</u><u> of the function that represents a parabola.</u>
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.
<u>2) Find the vertex of the first function, f(x) = x²</u>
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).
<u>3) Find teh vertex of the second function, g(x) = x² -10x + 2</u>
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.
