Pregunta
Toplic realize. Topiew (LMS graded)
Part 1 of 2
Find the zeros of the function, and then describe the behavior of the graph at each zero.
Part 1 of 2
Find the zeros of the function, and then describe the behavior of the graph at each zero.
Ask by Ingram Owen. in the United States
Dec 27,2024
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The zeros of the function
are
and
. At
, the graph crosses the x-axis, and at
, it touches the x-axis and turns around without crossing.
Solución
To find the zeros of the function
and describe the behavior of the graph at each zero, follow these steps:
1. Factor the Function
First, factor the given cubic function:
Factor out the greatest common factor (which is
):
Next, factor the quadratic expression
:
So, the fully factored form of the function is:
2. Find the Zeros
Set each factor equal to zero to find the zeros:
-
First Zero:
-
Second Zero:
Thus, the zeros of the function are:
3. Determine the Multiplicity of Each Zero
-
Zero at
: This zero comes from the factor and has a multiplicity of 1. -
Zero at
: This zero comes from the factor and has a multiplicity of 2.
4. Describe the Behavior at Each Zero
-
At
(Multiplicity 1): - Multiplicity is odd (1).
- Behavior: The graph crosses the x-axis at this point.
-
At
(Multiplicity 2): - Multiplicity is even (2).
- Behavior: The graph touches the x-axis and turns around at this point without crossing.
5. Summary
-
Zeros:
-
(Multiplicity 1) -
(Multiplicity 2)
-
-
Graph Behavior:
- At
: The graph crosses the x-axis. - At
: The graph touches the x-axis and turns around.
- At
6. Graphical Illustration
Here’s a rough sketch to visualize the behavior:
|
f(x)
|
*--•--*--•
|
-2 0
- The graph crosses the x-axis at
. - The graph touches and turns around at
.
This analysis fully describes the zeros and the corresponding behavior of the graph for the given function.
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Como
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Bonus Knowledge
To find the zeros of the function
, we can start by factoring out the greatest common factor, which is
:
Next, we can further factor the quadratic part:
Thus, our factored function is:
Setting the function to zero to find the zeros, we have:
From here, we can see that the zeros are:
-
-
(with multiplicity 2)
Now, let’s discuss the behavior of the graph at these zeros:
- At
, the graph crosses the x-axis since it’s a simple zero (multiplicity of 1). - At
, because of the multiplicity of 2, the graph touches the x-axis and bounces back up, rather than crossing over.
So, we have one crossing point at
and a bouncing point at
. Happy graphing!
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