Step 1:

Domain: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»|«/mi»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math».

Step 2:

Intercepts: Find both the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercepts.

Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»0«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», the graph crosses both axes at the origin.

Step 3:

Asymptotes: Check for any discontinuities in the trigonometric function.

The given function is continuous on the given interval. There are no asymptotes.

Step 4:

Symmetry: Test to see if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» or if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».

Since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» is an odd function and its graph will be symmetric about the origin.

Step 5:

Intervals of Increase and Decrease: Find the first derivative and the critical point(s) on the given interval of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math».

One endpoint of the given function occurs at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math». Since the derivative is zero at this point, the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/mstyle»«/math» cannot be included in the interval when checking for intervals of increase and decrease.

Set up a chart to summarize the information found.

Critical Points	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/mstyle»«/math»
Intervals	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math»	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#960;«/mo»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math»
Sign of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math»	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math»
Behaviour	increasing	increasing

The function is increasing on the intervals «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8746;«/mo»«/math» «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»(«/mi»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mfenced open=¨¨ close=¨]¨»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfenced»«/mstyle»«/math».

Step 6:

Local Extrema: Using the critical points and the intervals of increase and decrease obtained from the first derivative, find the local extrema.

Substitute the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinate of the critical points into the original function to find the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates.


The critical points are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math». However, since the function is increasing on the intervals «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»(«/mi»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mfenced open=¨¨ close=¨]¨»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfenced»«/mstyle»«/math», there are no local extrema.

Because the given function also has defined endpoints, the function values at the endpoints must be tested for the existence of absolute extrema.


Based on the coordinates of the critical points and the endpoints, there is an absolute maximum at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and there is an absolute minimum at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».

Step 7:

Concavity and possible inflection point(s): Use the second derivative to find possible inflection points and intervals of concavity.


Inflection points do not occur at the endpoints (concavity cannot change at an endpoint), so given the domain restriction of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math», the only possible inflection points are at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math».

To find the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates of the possible inflection points, substitute the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinates into the original function.


The possible inflection points are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi mathvariant=¨normal¨»§#960;«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfenced»«/mstyle»«/math».

The curve is concave up where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».


The curve is concave down on the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfenced»«/mstyle»«/math». The curve is concave up on the intervals «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mi mathvariant=¨normal¨»§#960;«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math» «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8746;«/mo»«/math» «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi mathvariant=¨normal¨»§#960;«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfenced»«/mstyle»«/math». Since there was a change in concavity at each of the possible inflection points, and the function is continuous, both points are considered inflection points. Therefore, the inflection points are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi mathvariant=¨normal¨»§#960;«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»§#960;«/mi»«/mrow»«/mfenced»«/mstyle»«/math».

Step 8:

Sketch: Using the information obtained in Steps 1 to 7, sketch the graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».

Summary
domain	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
intercepts	«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
asymptotes	none
symmetry	odd; symmetric about origin
intervals of increase and decrease and critical points	increasing: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#960;«/mo»«mo»,«/mo»«mfenced open=¨¨ close=¨]¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» CP: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
local and absolute extrema	no local extrema absolute max: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» absolute min: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»
concavity and inflection points	down: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» up: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» IP: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math»