Question

the graph to the right was created by using one horizontal shift and one vertical shift of the graph of the basic function y = x^3. write a function that describes the graph to the right.
the function that describes the graph is f(x) = (simplify your answer.)

Explanation:

Step1: Recall shift - rules

For a function $y = f(x)$, a horizontal shift of $h$ units and a vertical shift of $k$ units gives $y=f(x - h)+k$.

Step2: Identify the horizontal shift

Let's assume the horizontal shift is $h$ units. If the graph of $y = x^{3}$ is shifted $h$ units to the right, the function becomes $y=(x - h)^{3}$. If shifted to the left, $h$ is negative.

Step3: Identify the vertical shift

Let the vertical shift be $k$ units. The function then becomes $y=(x - h)^{3}+k$. Suppose by observing the graph (not shown in detail here but in general), if the vertex of $y = x^{3}$ which is at $(0,0)$ has moved to $(a,b)$, then $h=a$ and $k = b$. For example, if the graph of $y=x^{3}$ is shifted 2 units to the left and 3 units up, $h=- 2$ and $k = 3$, and the function is $y=(x+2)^{3}+3$.

Answer:

Assume the horizontal shift is $h$ units and the vertical shift is $k$ units, the function is $f(x)=(x - h)^{3}+k$ (You need to determine $h$ and $k$ by observing the specific graph, for example, if the graph is shifted 2 units to the left and 4 units up, $f(x)=(x + 2)^{3}+4$)