consider the following graph of $g(x)$. write a formula for $g(x)$ that describes the transformations…

Explanation:

Step1: Identify the parent function

The graph shows a cubic root shape, so the parent function is $f(x) = \sqrt[3]{x}$.

Step2: Determine the horizontal shift

The inflection point of the parent function $(0,0)$ has moved to $x = 5$. $$h = 5$$

Step3: Determine the vertical shift

The inflection point of the parent function $(0,0)$ has moved to $y = 4$. $$k = 4$$

Step4: Determine the vertical stretch or compression

Identify another point on the graph, such as $(6, 5)$. Substitute into $g(x) = a\sqrt[3]{x - h} + k$. $$5 = a\sqrt[3]{6 - 5} + 4$$ $$5 = a(1) + 4 \implies a = 1$$

Step5: Write the final formula

Combine the identified transformations into the general function form. $$g(x) = \sqrt[3]{x - 5} + 4$$

Answer:

$g(x) = \sqrt[3]{x - 5} + 4$