Question

shifting the cube root function quick check
analyze the effect on the x-intercept of the graph of $f(x) = \sqrt3{x}$ when $f(x)$ is replaced by $f(x + c)$ and $c$ is positive. (1 point)
\bigcirc the x-intercept shifts up by a distance of $c$
\bigcirc the x-intercept shifts down by a distance of $c$
\bigcirc the x-intercept shifts to the right by a distance of $c$
\bigcirc the x-intercept shifts left by a distance of $c$

Explanation:

Step1: Find x-intercept of \( f(x)=\sqrt[3]{x} \)

To find the x - intercept, set \( f(x) = 0 \). So, \( \sqrt[3]{x}=0 \), which implies \( x = 0 \). The x - intercept of \( f(x)=\sqrt[3]{x} \) is \( (0,0) \).

Step2: Find x-intercept of \( f(x + c)=\sqrt[3]{x + c} \)

Set \( f(x + c)=0 \), then \( \sqrt[3]{x + c}=0 \). Cubing both sides, we get \( x + c=0 \), so \( x=-c \). The x - intercept of \( f(x + c) \) is \( (-c,0) \).

Step3: Analyze the shift

The original x - intercept was at \( x = 0 \), and the new x - intercept is at \( x=-c \) (where \( c>0 \)). Moving from \( x = 0 \) to \( x=-c \) means the graph shifts to the left by a distance of \( c \) units (since we are moving from 0 to a negative number with magnitude \( c \), which is a leftward shift on the x - axis).

Answer:

The x - intercept shifts left by a distance of \( c \) (the option: The x - intercept shifts left by a distance of \( c \))