Question

how does the graph of $f(x) = \sqrt3{x}$ change when $f(x)$ is replaced by $f(x) + d$ when $d$ is positive? (1 point) \
\bigcirc the graph shifts up. \
\bigcirc the graph shifts down. \
\bigcirc the graph shifts to the left. \
\bigcirc the graph shifts to the right.

Explanation:

To determine how the graph of \( f(x)=\sqrt[3]{x} \) changes when replaced by \( f(x)+d \) (where \( d>0 \)), we use the rules of function transformations. For a function \( y = f(x) \), the transformation \( y = f(x)+k \) (where \( k \) is a constant) causes a vertical shift. If \( k>0 \), the graph shifts up by \( k \) units; if \( k<0 \), it shifts down. Here, \( k = d \) and \( d>0 \), so the graph of \( f(x)+d \) is the graph of \( f(x) \) shifted up by \( d \) units. Horizontal shifts are caused by transformations like \( f(x + h) \) (shift left if \( h>0 \)) or \( f(x - h) \) (shift right if \( h>0 \)), which is not the case here.

Answer:

The graph shifts up.