Pregunta
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Consider the following polynomial function.
Answer the questions regarding the graph of .
Then, use this information to graph the function.
(d) Graph by doing the following.
  • Plot all points where the graph of intersects the -axis or -axis.
  • For each point on the -axis, select the correct behavior.
  • Click on the graph icon.

Ask by Warren Schneider. in the United States
Mar 12,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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(a) End Behavior
  • As approaches negative infinity, approaches negative infinity.
  • As approaches positive infinity, approaches positive infinity.
(b) Real Zeros
  • (graph touches the -axis)
  • (graph crosses the -axis)
© -Intercept
(d) Graphing the Function
  1. Plot the -intercepts at and .
  2. Plot the -intercept at .
  3. Sketch the curve following the end behaviors:
    • Starts in the third quadrant.
    • Touches the -axis at without crossing.
    • Passes through .
    • Crosses the -axis at .
  4. Ensure the graph reflects the end behaviors as approaches positive and negative infinity.

Solución

(a) End Behavior
Since
expands to a cubic polynomial with a positive leading coefficient, the end behavior is:
  • As ,
  • As ,

(b) Real Zeros and Their Behavior
Setting the function equal to zero gives:
The zeros are:
  • (with multiplicity 2): The graph touches the -axis and turns around.
  • (with multiplicity 1): The graph crosses the -axis.
Thus:
  • Zero(s) where the graph touches, but does not cross the -axis:
  • Zero(s) where the graph crosses the -axis:

© -Intercept
To find the -intercept, substitute into :
So, the -intercept is .

(d) Graphing the Function
  1. Plot the Intercepts:
    • -intercepts: and
      • At , the graph touches the -axis.
      • At , the graph crosses the -axis.
    • -intercept:
  2. Plot Additional Points (if needed):
    You can calculate a few extra points on either side of the intercepts to better capture the curve.
  3. Indicate the End Behavior:
    • As : the graph goes down (toward ).
    • As : the graph goes up (toward ).
  4. Sketch the Graph:
    • Begin in the third quadrant following the left end behavior.
    • Rise to touch the -axis at (without crossing), then descend to the -intercept .
    • Finally, rise again and cross the -axis at following the right end behavior.
By following these steps, the graph of is fully determined.

Respondido por UpStudy AI y revisado por un tutor profesional

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Mind Expander

The polynomial function has an end behavior that reflects its leading term, which is . As , ; and as , . This shows the familiar “up-right” and “down-left” behavior typical of cubic functions.
Next, for the real zeros of , we find with a multiplicity of 2, meaning the graph touches but does not cross the x-axis at this point. The other zero is , where the graph crosses the x-axis. Therefore, the zeros are: (touches) and (crosses). The y-intercept is calculated by evaluating . So, the y-intercept is .

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