Question

fill in each blank with the appropriate response (remember that the vertical stretch or shrink factor is positive ) the graph of ( y = 5sqrt3{x + 4} ) can be obtained from the graph of ( y = sqrt3{x} ) by shifting horizontally _ units to the _ and stretching vertically by applying a factor of ____. the graph of ( y = 5sqrt3{x + 4} ) can be obtained from the graph of ( y = sqrt3{x} ) by shifting horizontally \\(\square\\) units to the \\(\blacktriangledown\\) and stretching vertically by applying a factor of \\(\square\\)

Explanation:

Step1: Analyze horizontal shift

For a function \( y = \sqrt[3]{x + h} \), the horizontal shift is \( h \) units. If \( h>0 \), it shifts left; if \( h<0 \), it shifts right. Here, the function is \( y = 5\sqrt[3]{x + 4} \), so comparing to \( y=\sqrt[3]{x} \), \( h = 4 \), so it shifts 4 units to the left.

Step2: Analyze vertical stretch factor

For a function \( y = a\sqrt[3]{x} \), the vertical stretch factor is \( |a| \). Here, \( a = 5 \), so the vertical stretch factor is 5.

Answer:

The graph of \( y = 5\sqrt[3]{x + 4} \) can be obtained from the graph of \( y=\sqrt[3]{x} \) by shifting horizontally \( \boldsymbol{4} \) units to the \( \boldsymbol{\text{left}} \) and stretching vertically by applying a factor of \( \boldsymbol{5} \).