Question

F(x) = x^2. What is g(x)?

Answer 1

`\textsf{B.} \quad g(x)=-x^2-2`

Explanation:

Transformations

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}.`

`f(x)-a \implies f(x) \: \textsf{translated $a$ units down}.`

`-f(x) \implies f(x) \: \textsf{reflected in the $x$-axis}.`

`f(-x) \implies f(x) \: \textsf{reflected in the $y$-axis}.`

From inspection of the given graph, we can see that g(x) is function f(x) reflected in the x-axis and translated 2 units down.

Given function f(x):

`f(x)=x^2`

Reflected in the x-axis:

`-f(x) \implies g(x)= -x^2`

Translated 2 units down:

`-f(x)-2 \implies g(x)=-x^2-2`

Therefore:

`g(x)=-x^2-2`

Note:

If the leading coefficient of a quadratic function is positive, the parabola opens upwards.

If the leading coefficient of a quadratic function is negative, the parabola opens downwards.

Therefore, as g(x) opens downwards, the term in x² will be negative.