Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis.
-$$f ( x ) = ( x - 2 ) ^ { 2 } ( x + 4 ) ^ { 2 }$$
A) above the $x$-axis: $( - 4,2 )$
below the x-axis: $( - \infty , - 4 ) , ( 2 , \infty )$

B) above the $x$-axis: no intervals
below the x-axis: $( - \infty , - 4 ) , ( - 4,2 ) , ( 2 , \infty )$

C) above the $x$-axis: $( - \infty , - 4 ) , ( - 4,2 ) , ( 2 , \infty )$
below the $x$-axis: no intervals

D) above the $x$-axis: $( - \infty , - 4 ) , ( 2 , \infty )$
below the x-axis: $( - 4,2 )$

Question

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis.
-$$f ( x ) = ( x - 2 ) ^ { 2 } ( x + 4 ) ^ { 2 }$$
A) above the $x$-axis: $( - 4,2 )$
below the x-axis: $( - \infty , - 4 ) , ( 2 , \infty )$

B) above the $x$-axis: no intervals
below the x-axis: $( - \infty , - 4 ) , ( - 4,2 ) , ( 2 , \infty )$

C) above the $x$-axis: $( - \infty , - 4 ) , ( - 4,2 ) , ( 2 , \infty )$
below the $x$-axis: no intervals

D) above the $x$-axis: $( - \infty , - 4 ) , ( 2 , \infty )$
below the x-axis: $( - 4,2 )$

Quizplus · Accepted Answer

The function $f(x) = (x - 2)^2(x + 4)^2$ has x-intercepts at $x = 2$ and $x = -4$, both of which are points where the graph touches the x-axis but does not cross it, because the factors are squared. This means the function does not change signs at these intercepts and remains positive on both sides of each intercept. Therefore, the graph of $f$ is above the x-axis for all values of $x$, making option C correct.

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