$f(x)=(x+1)(x-2)(x-3)^2$
For the single roots -1 and 2, the graph
$\square$ the $x$-axis at the intercepts.
For the double root 3, the graph
$\square$ the $x$-axis at the intercepts.
Answer (1)
For single roots -1 and 2, the graph crosses the x-axis.
For the double root 3, the graph touches the x-axis.
Single roots cause the function to change sign, leading to the graph crossing the x-axis.
Double roots cause the function to maintain its sign, leading to the graph touching the x-axis.
The graph crosses at single roots and touches at double roots, so the answer is crosses , t o u c h es .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 1 ) ( x − 2 ) ( x − 3 ) 2 . We need to describe the behavior of the graph at the x-axis intercepts for single and double roots.
Identifying the Roots The roots of the function are -1, 2, and 3. The roots -1 and 2 are single roots, meaning the factor corresponding to these roots appears with a power of 1. The root 3 is a double root, meaning the factor corresponding to this root appears with a power of 2.
Behavior at Single Roots For single roots, the graph crosses the x-axis. This is because the function changes sign at these roots. For example, near x = − 1 , the function behaves like ( x + 1 ) , which changes from negative to positive as x passes through -1. Similarly, near x = 2 , the function behaves like ( x − 2 ) , which changes from negative to positive as x passes through 2.
Behavior at Double Roots For double roots, the graph touches the x-axis and turns around (it does not cross the x-axis). This is because the function does not change sign at these roots. For example, near x = 3 , the function behaves like ( x − 3 ) 2 , which is always non-negative. Therefore, the graph touches the x-axis at x = 3 and bounces back.
Final Answer Therefore, for the single roots -1 and 2, the graph crosses the x-axis at the intercepts. For the double root 3, the graph touches the x-axis at the intercepts.
Examples
Understanding the behavior of polynomial functions at their roots is crucial in various fields. For instance, in physics, when modeling oscillations or waves, single roots represent points where the wave crosses the equilibrium, while double roots indicate points where the wave momentarily touches the equilibrium before reversing direction. In engineering, analyzing the stability of systems often involves examining the roots of characteristic equations, where single and double roots have distinct implications for system response.
