Question

The dotted graph is a reflection of the solid graph. The solid graph is a cubic function with the equation  f(x)=x3+1 . Enter the option number that shows the equation of the image.Option #1:  g(x)=−x2+1 Option #2:  g(x)=−x3−1 Option #3:  g(x)=−x3+1 Option #4:  g(x)=x3−1

Answer 1

To find the equation of the dotted graph, which is a reflection of the solid graph (a cubic function), we first need to understand how reflections work.

The equation of the solid graph is given as:
\[ f(x) = x^3 + 1. \]

When we reflect a graph across the x-axis, the transformation applies to the output of the function. This means we replace \( f(x) \) with \( -f(x) \).

So, reflecting \( f(x) \):
\[
g(x) = -f(x) = - (x^3 + 1) = -x^3 - 1.
\]

Now, among the options provided, we look for a match with our derived equation \( g(x) = -x^3 - 1 \).

- Option #1: \( g(x) = -x^2 + 1 \) (not a match)
- Option #2: \( g(x) = -x^3 - 1 \) (this is a match)
- Option #3: \( g(x) = -x^3 + 1 \) (not a match)
- Option #4: \( g(x) = x^3 - 1 \) (not a match)

Therefore, the correct option is **Option #2: \( g(x) = -x^3 - 1 \)**.