Question 1205524
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The idea is to plot a bunch of (x,y) points to draw a curve through them. The more points, the more accurate the curve.


To generate any given point, plug in some value for x to find y.


Let's say we did x = 0.
y=x^3-5x+1
y=0^3-5*0+1
y=0-0+1
y=1
The point (0,1) is on this curve.


Now try x = 1
y=x^3-5x+1
y=1^3-5*1+1
y=1-5+1
y=-3
The point (1,-3) is on this curve.


Repeat the process for other x values to generate a table such as this
<table border = "1" cellpadding = "5"><tr><td>x</td><td>y</td></tr><tr><td>-2</td><td>3</td></tr><tr><td>-1</td><td>5</td></tr><tr><td>0</td><td>1</td></tr><tr><td>1</td><td>-3</td></tr><tr><td>2</td><td>-1</td></tr></table>
Feel free to try other x values. 


Once you have your (x,y) points, plot them on the same xy grid and draw a curve through them. 
{{{
drawing(400,400,-5,5,-4,6,
graph(400,400,-5,5,-4,6,-100,x^3-5x+1),
circle(-2,3,0.05),circle(-2,3,0.07),circle(-2,3,0.09),circle(-2,3,0.11),circle(-2,3,0.13),circle(-2,3,0.15),circle(-1,5,0.05),circle(-1,5,0.07),circle(-1,5,0.09),circle(-1,5,0.11),circle(-1,5,0.13),circle(-1,5,0.15),circle(0,1,0.05),circle(0,1,0.07),circle(0,1,0.09),circle(0,1,0.11),circle(0,1,0.13),circle(0,1,0.15),circle(1,-3,0.05),circle(1,-3,0.07),circle(1,-3,0.09),circle(1,-3,0.11),circle(1,-3,0.13),circle(1,-3,0.15),circle(2,-1,0.05),circle(2,-1,0.07),circle(2,-1,0.09),circle(2,-1,0.11),circle(2,-1,0.13),circle(2,-1,0.15)
)
}}}
Keep in mind that I'm using technology to plot the green curve shown above. For now the only real way to know when and where the curve changes direction is to plot more points to see a more accurate version of the curve.
Later on you can use Calculus to pinpoint the local min and local max.


GeoGebra and Desmos are two of many graphing tools that you can use for verifying the answer. 


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If you had a table of points such as this
<table border = "1" cellpadding = "5"><tr><td>x</td><td>y</td></tr><tr><td>-2</td><td>3</td></tr><tr><td>-1.5</td><td>5.125</td></tr><tr><td>-1</td><td>5</td></tr><tr><td>-0.5</td><td>3.375</td></tr><tr><td>0</td><td>1</td></tr><tr><td>0.5</td><td>-1.375</td></tr><tr><td>1</td><td>-3</td></tr><tr><td>1.5</td><td>-3.125</td></tr><tr><td>2</td><td>-1</td></tr></table>
then the graph would be
{{{
drawing(400,400,-5,5,-4,6,
graph(400,400,-5,5,-4,6,-100,x^3-5x+1),
circle(-2,3,0.05),circle(-2,3,0.07),circle(-2,3,0.09),circle(-2,3,0.11),circle(-2,3,0.13),circle(-2,3,0.15),circle(-1.5,5.125,0.05),circle(-1.5,5.125,0.07),circle(-1.5,5.125,0.09),circle(-1.5,5.125,0.11),circle(-1.5,5.125,0.13),circle(-1.5,5.125,0.15),circle(-1,5,0.05),circle(-1,5,0.07),circle(-1,5,0.09),circle(-1,5,0.11),circle(-1,5,0.13),circle(-1,5,0.15),circle(-0.5,3.375,0.05),circle(-0.5,3.375,0.07),circle(-0.5,3.375,0.09),circle(-0.5,3.375,0.11),circle(-0.5,3.375,0.13),circle(-0.5,3.375,0.15),circle(0,1,0.05),circle(0,1,0.07),circle(0,1,0.09),circle(0,1,0.11),circle(0,1,0.13),circle(0,1,0.15),circle(0.5,-1.375,0.05),circle(0.5,-1.375,0.07),circle(0.5,-1.375,0.09),circle(0.5,-1.375,0.11),circle(0.5,-1.375,0.13),circle(0.5,-1.375,0.15),circle(1,-3,0.05),circle(1,-3,0.07),circle(1,-3,0.09),circle(1,-3,0.11),circle(1,-3,0.13),circle(1,-3,0.15),circle(1.5,-3.125,0.05),circle(1.5,-3.125,0.07),circle(1.5,-3.125,0.09),circle(1.5,-3.125,0.11),circle(1.5,-3.125,0.13),circle(1.5,-3.125,0.15),circle(2,-1,0.05),circle(2,-1,0.07),circle(2,-1,0.09),circle(2,-1,0.11),circle(2,-1,0.13),circle(2,-1,0.15)
)
}}}
which is a slightly more accurate look. 
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