Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis.
-$$f ( x ) = ( x - 3 ) ^ { 3 }$$
A) above the $x$-axis: $( 3 , \infty )$
below the x-axis: $( - \infty , 3 )$

B) above the $x$-axis: $( - \infty , 3 )$
below the $x$-axis: $( 3 , \infty )$

C) above the $x$-axis: no intervals
below the $x$-axis: $( - \infty , 3 ) , ( 3 , \infty )$

D) above the $x$-axis: $( - \infty , 3 ) , ( 3 , \infty )$
below the x-axis: no intervals

Question

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis.
-$$f ( x ) = ( x - 3 ) ^ { 3 }$$
A) above the $x$-axis: $( 3 , \infty )$
below the x-axis: $( - \infty , 3 )$

B) above the $x$-axis: $( - \infty , 3 )$
below the $x$-axis: $( 3 , \infty )$

C) above the $x$-axis: no intervals
below the $x$-axis: $( - \infty , 3 ) , ( 3 , \infty )$

D) above the $x$-axis: $( - \infty , 3 ) , ( 3 , \infty )$
below the x-axis: no intervals

Quizplus · Accepted Answer

The function $f(x) = (x - 3)^3$ has a single x-intercept at $x = 3$. Since the exponent is odd, the function changes sign at this intercept. Given the positive coefficient of the leading term, the function is negative for $x < 3$ and positive for $x > 3$, indicating it is below the x-axis for $(- \infty, 3)$ and above the x-axis for $(3, \infty)$.

Use the X-Intercepts to Find the Intervals on Which the Graph