Question

f(x) = \sqrt3{x}\
g(x) = \sqrt3{x} + k\
find the value of k .\
k = \square\
compare the graphs.\
the graph of g is a \underline{\qquad} of the graph of f .\
\
\\ \\ translation 4 units left\\ \\ \\ \\ translation 4 units right\\ \\ \\ \\ translation 4 units up\\ \\ \\ \\ translation 4 units down

Explanation:

Step1: Identify a point on \( f(x) \) and \( g(x) \)

The parent function is \( f(x) = \sqrt[3]{x} \). At \( x = 0 \), \( f(0) = \sqrt[3]{0} = 0 \), so the point \( (0, 0) \) is on \( f(x) \). For \( g(x) = \sqrt[3]{x} + k \), when \( x = 0 \), \( g(0) = \sqrt[3]{0} + k = k \). From the graph, we can see that \( g(x) \) passes through \( (0, 4) \), so when \( x = 0 \), \( g(0) = 4 \).

Step2: Solve for \( k \)

Since \( g(0) = k \) and \( g(0) = 4 \), we have \( k = 4 \).

Step3: Determine the transformation

The function \( g(x) = f(x) + 4 \). In function transformations, adding a constant to the function \( f(x) \) (i.e., \( f(x) + k \) with \( k>0 \)) results in a vertical translation up by \( k \) units. Here, \( k = 4 \), so the graph of \( g(x) \) is a translation of the graph of \( f(x) \) 4 units up.

Answer:

For the value of \( k \): \( k = 4 \)
For the transformation: translation 4 units up