Pregunta
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.
Then, use this information to graph the function.
Ask by Warren Schneider. in the United States
Mar 12,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
(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
- Plot the
-intercepts at and . - Plot the
-intercept at . - Sketch the curve following the end behaviors:
- Starts in the third quadrant.
- Touches the
-axis at without crossing. - Passes through
. - Crosses the
-axis at .
- 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
-
Plot the Intercepts:
-
-intercepts: and - At
, the graph touches the -axis. - At
, the graph crosses the -axis.
- At
-
-intercept:
-
-
Plot Additional Points (if needed):
You can calculate a few extra points on either side of the intercepts to better capture the curve. -
Indicate the End Behavior:
- As
: the graph goes down (toward ). - As
: the graph goes up (toward ).
- As
-
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
Como
error msg


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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