Question

Which translation maps the graph of the function f(x) = x2 onto the function g(x) = x2 + 2x + 6?

Answer 1

Answer:

The translation is left 1 unit, up 5 units ⇒ A

Step-by-step explanation:

Let us revise the rules of translation of a function

• If the function f(x) translated horizontally to the right by h units, then its image is g(x) = f(x - h)  

• If the function f(x) translated horizontally to the left by h units, then its image is g(x) = f(x + h)  

• If the function f(x) translated vertically up by k units, then its image is g(x) = f(x) + k  

• If the function f(x) translated vertically down by k units, then its image is g(x) = f(x) - k

The vertex form of the quadratic function is f(x) = ax² + bx + c, where a, b, and c are constant is f(x) = a(x - h)² + k, where h = $\frac{-b}{2a}$  and k = f(h)

You will use the vertex form to find the translation

∵ g(x) = x² + 2x + 6

∴ a = 1 and b = 2

→ use the rule of h to find it

∴ h = $\frac{-2}{2(1)}=\frac{-2}{2}$

∴ h = -1

→ Use it to find k

∵ g(h) = k

∴ k = g(-1)

→ Substitute x by -1 in g to find k

∴ k = (-1)² + 2(-1) + 6 = 1 - 2 + 6

∴ k = 5

→ Substitute the values of h and k in the vertex form above

∴ g(x) = 1(x - -1)² + 5

∴ g(x) = (x + 1)² + 5

→ By using the 2nd and 3rd rules of translation above

∴ f(x) is translated 1 unit to the left and 5 units up

∴ The translation is left 1 unit, up 5 units