Answer:
The new function is (x + 7)² - 47 ⇒ the 3rd answer
Step-by-step explanation:
* Lets put the function f(x) in the vertex form at first and then make
the translation
∵ The general form of the quadratic function is
f(x) = ax² + bx + c
∵ The x-coordinate of the vertex of the function is -b/2a
∵ The y-coordinate of the vertex of the function is f(-b/2a)
- Lets find a , b from the function two find the vertex point
∵ f(x) = x² + 22x + 58
∴ a = 1 , b = 22 , c = 58
∵ x-coordinate of the vertex = -b/2a
∴ x-coordinate of the vertex = -22/2(1) = -11
∵ y-coordinate of the vertex = f(-11)
∴ f(-11) = (-11)² + 22(-11) + 58 = 121 - 242 + 58 = -63
∴ The vertex point is (-11 , -63)
- The vertex form of the quadratic function is f(x) = (x - h)² + k , where
(h , k) are the coordinates of the vertex point
∵ The vertex point is (-11 , -63)
∴ h = -11 , k = -63
∴ f(x) = (x - -11)² + -63
∴ f(x) = (x + 11)² - 63
* lets revise the rules of the translation
- If the function f(x) translated horizontally to the right
by m units, then the new function g(x) = f(x - m)
- If the function f(x) translated horizontally to the left
by m units, then the new function g(x) = f(x + m)
- If the function f(x) translated vertically up
by n units, then the new function g(x) = f(x) + n
- If the function f(x) translated vertically down
by n units, then the new function g(x) = f(x) – n
∵ f(x) will translate 4 units to the right
∴ m = 4
∵ f(x) ⇒ f(x - m)
∴ (x + 11)² ⇒ (x + 11 - 4)² = (x + 7)²
∵f(x) will translate 16 units up
∴ -63 will add by 16
∴ n = 16
∴ f(x) ⇒ f(x) + n
∵ -63 + 16 = -47
∴ The new function is (x + 7)² - 47
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