Sketch the graph of the function $P ( x ) = ( x - 1 ) ^ { 4 } + 16$ by transforming the graph of an appropriate function of the form $y = x ^ { n }$

Question

Quizplus · Accepted Answer

The function $P(x)=(x-1)^4+16$ can be rewritten as:
$$P(x)=(x-1)^4+16=x^4-4x^3+6x^2-4x+1+16$$
$$\Rightarrow P(x)=x^4-4x^3+6x^2-4x+17$$
Now, the function $y=x^4$ has a similar shape to the graph of $P(x)$, with the vertex at the origin. To obtain the vertex of the graph of $P(x)$, we need to shift the vertex of the graph of $y=x^4$ to the right by 1 and up by 16. Therefore, the graph of $P(x)$ can be obtained by applying the following transformations to the graph of $y=x^4$:
$$y=x^4\to y=(x-1)^4 \to y=(x-1)^4+16$$
This corresponds to shifting the graph of $y=x^4$ to the right by 1 unit and up by 16 units. The correct graph is shown in choice (C).

Sketch the Graph of the Function P(x)=(x−1)4+16P ( x ) = ( x - 1 ) ^ { 4 } + 16P(x)=(x−1)4+16

Sketch the Graph of the Function $P ( x ) = ( x - 1 ) ^ { 4 } + 16$