Question 116779
To graph {{{y=x^2+5x+3}}} simply plot a sequence of points and connect them. In other words, find y values for some given x values and plot the points (x,y).




So let's start at x=-6


Let's evaluate {{{f(-6)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(-6)=(-6)^2+5(-6)+3}}} Plug in {{{x=-6}}}. In other words, replace each x with -6.



{{{f(-6)=(36)+5(-6)+3}}} Evaluate {{{(-6)^2}}} to get 36.



{{{f(-6)=(36)+-30+3}}} Multiply 5 and -6 to get  -30



{{{f(-6)=9}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(-5)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(-5)=(-5)^2+5(-5)+3}}} Plug in {{{x=-5}}}. In other words, replace each x with -5.



{{{f(-5)=(25)+5(-5)+3}}} Evaluate {{{(-5)^2}}} to get 25.



{{{f(-5)=(25)+-25+3}}} Multiply 5 and -5 to get  -25



{{{f(-5)=3}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(-4)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(-4)=(-4)^2+5(-4)+3}}} Plug in {{{x=-4}}}. In other words, replace each x with -4.



{{{f(-4)=(16)+5(-4)+3}}} Evaluate {{{(-4)^2}}} to get 16.



{{{f(-4)=(16)+-20+3}}} Multiply 5 and -4 to get  -20



{{{f(-4)=-1}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(-3)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(-3)=(-3)^2+5(-3)+3}}} Plug in {{{x=-3}}}. In other words, replace each x with -3.



{{{f(-3)=(9)+5(-3)+3}}} Evaluate {{{(-3)^2}}} to get 9.



{{{f(-3)=(9)+-15+3}}} Multiply 5 and -3 to get  -15



{{{f(-3)=-3}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(-2)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(-2)=(-2)^2+5(-2)+3}}} Plug in {{{x=-2}}}. In other words, replace each x with -2.



{{{f(-2)=(4)+5(-2)+3}}} Evaluate {{{(-2)^2}}} to get 4.



{{{f(-2)=(4)+-10+3}}} Multiply 5 and -2 to get  -10



{{{f(-2)=-3}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(-1)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(-1)=(-1)^2+5(-1)+3}}} Plug in {{{x=-1}}}. In other words, replace each x with -1.



{{{f(-1)=(1)+5(-1)+3}}} Evaluate {{{(-1)^2}}} to get 1.



{{{f(-1)=(1)+-5+3}}} Multiply 5 and -1 to get  -5



{{{f(-1)=-1}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(0)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(0)=(0)^2+5(0)+3}}} Plug in {{{x=0}}}. In other words, replace each x with 0.



{{{f(0)=(0)+5(0)+3}}} Evaluate {{{(0)^2}}} to get 0.



{{{f(0)=(0)+0+3}}} Multiply 5 and 0 to get  0



{{{f(0)=3}}} Now combine like terms



-----------Now let's evaluate another value---------



Let's evaluate {{{f(1)}}}



{{{f(x)=x^2+5x+3}}} Start with the given function.



{{{f(1)=(1)^2+5(1)+3}}} Plug in {{{x=1}}}. In other words, replace each x with 1.



{{{f(1)=(1)+5(1)+3}}} Evaluate {{{(1)^2}}} to get 1.



{{{f(1)=(1)+5+3}}} Multiply 5 and 1 to get  5



{{{f(1)=9}}} Now combine like terms



Now lets make a table of the values we have calculated

<pre>
<TABLE width=500>

<TR><TD> x</TD><TD>y</TD></TR>

<TR><TD> -6</TD><TD>9</TD></TR> 
<TR><TD> -5</TD><TD>3</TD></TR> 
<TR><TD> -4</TD><TD>-1</TD></TR> 
<TR><TD> -3</TD><TD>-3</TD></TR> 
<TR><TD> -2</TD><TD>-3</TD></TR> 
<TR><TD> -1</TD><TD>-1</TD></TR> 
<TR><TD> 0</TD><TD>3</TD></TR> 
<TR><TD> 1</TD><TD>9</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-10,10,-10,10,
  grid( 1 ),
circle(-6,9,0.05),
circle(-6,9,0.08),
circle(-5,3,0.05),
circle(-5,3,0.08),
circle(-4,-1,0.05),
circle(-4,-1,0.08),
circle(-3,-3,0.05),
circle(-3,-3,0.08),
circle(-2,-3,0.05),
circle(-2,-3,0.08),
circle(-1,-1,0.05),
circle(-1,-1,0.08),
circle(0,3,0.05),
circle(0,3,0.08),
circle(1,9,0.05),
circle(1,9,0.08))}}}



Now connect the points to graph {{{y=x^2+5x+3}}}  (note: the more points you plot, the easier it is to draw the graph)

{{{drawing(900,900,-10,10,-10,10,
grid( 1 ),
graph(900,900,-10,10,-10,10, x^2+5x+3),
circle(-6,9,0.05),
circle(-6,9,0.08),
circle(-5,3,0.05),
circle(-5,3,0.08),
circle(-4,-1,0.05),
circle(-4,-1,0.08),
circle(-3,-3,0.05),
circle(-3,-3,0.08),
circle(-2,-3,0.05),
circle(-2,-3,0.08),
circle(-1,-1,0.05),
circle(-1,-1,0.08),
circle(0,3,0.05),
circle(0,3,0.08),
circle(1,9,0.05),
circle(1,9,0.08))}}}