Question

The point of inflection of the cubic function f(x)
is (−5,0)
. What is the equation of the new function h(x)
that is obtained by shifting f(x)
6 units to the right?(1 point)
Responses

h(x)=(x−11)3
h left parenthesis x right parenthesis equals left parenthesis x minus 11 right parenthesi cubed

h(x)=(x+11)3
h left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesi cubed

h(x)=(x+1)3
h left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesi cubed

h(x)=(x−1)3

Answer 1

To find the new function \( h(x) \) obtained by shifting the function \( f(x) \) 6 units to the right, we will use the transformation rule for functions. When a function \( f(x) \) is shifted \( c \) units to the right, the new function can be expressed as \( f(x - c) \).

In this case, we need to shift the function 6 units to the right:

\[
h(x) = f(x - 6)
\]

Given that the point of inflection of \( f(x) \) is \( (-5, 0) \), it indicates the behavior of \( f(x) \) but does not directly provide us with the equation of \( f(x) \). However, we only need to shift the previously defined cubic function.

For the original function to pass through the point \((-5, 0)\), if we consider a simple cubic function around this point, we could express it in the form:

\[
f(x) = k(x + 5)^3
\]

Then, shifting \( f(x) \) six units to the right would yield:

\[
h(x) = k((x - 6) + 5)^3 = k(x - 1)^3
\]

The value of \( k \) does not affect the position of the point of inflection. Therefore, the function can take the form \( h(x) = k(x - 1)^3 \).

From the possible answers listed, the correct one corresponds to the shifting as described:

\[
h(x) = (x - 1)^3
\]

Thus, the answer is:

**h(x) = (x - 1)³**