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) \( h(x)=(x-11)^{3} \) \( h(x)=(x-1)^{3} \) \( h(x)=(x+11)^{3} \) \( h(x)=(x+1)^{3} \)
Ask by Cook Davison. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( h(x) = (x - 1)^3 \)
Solution
To shift the cubic function \( f(x) \) six units to the right, you replace \( x \) with \( x - 6 \) in the original function.
Given that the inflection point of \( f(x) \) is \((-5, 0)\), the original function can be expressed as:
\[ f(x) = (x + 5)^3 \]
Shifting it six units to the right:
\[ h(x) = f(x - 6) = (x - 6 + 5)^3 = (x - 1)^3 \]
**Answer:**
\( h(x)=(x-1)^{3} \)
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Simplify this solution Extra Insights
When a cubic function has a point of inflection at \((-5, 0)\), the function can be expressed in the form \( f(x) \). To shift this function 6 units to the right, you simply replace \(x\) with \(x - 6\). Thus, the new function becomes \( h(x) = f(x - 6) \). Given the point of inflection of \( f(x) \), moving it 6 units to the right results in a new point of inflection at \((-5 + 6, 0) = (1, 0)\). Therefore, \( h(x) \) can be expressed as \( h(x) = (x - 1)^3 \), making the correct choice \( h(x) = (x - 1)^3 \).
