Question 177829
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a){{{y=x^2+4x+10}}}
Completing the square:
{{{y=(x^2+4x+4)+10-4}}}
{{{highlight(y=(x+2)^2+6)}}}
Vertex: (-2,6)
Axis of Symmetry=-2
Solving the X-intercepts:
{{{x=(-b+-sqrt(b^2-4ac))/(2a)}}}--->where{{{system(a=1,b=4,c=10)}}}
Solving the Discriminant: {{{b^2-4ac=4^2-4*1*10=16-40=-24}}}---> since negative, it has no solutions for real numbers.
Y-Intercept: {{{f(x)=0}}}
{{{y=0^2+4(0)+10}}}
{{{y=10}}}
Continuing,
{{{x=(-4+-i*sqrt(-24))/(2*1)=(-4+-i*4.8989795)/2}}}
Or see the graph,
{{{drawing(400,400,-8,5,-5,20,grid(1),graph(400,400,-8,5,-5,20,x^2+4x+10),green(line(-2,20,-2,-5)),circle(-2,6,.15),circle(0,10,.12))}}}
b){{{y=-x^2+6x-8}}}----->{{{y=-1(x^2+6x-8)}}}
Completing the square:
{{{y=-1(x^2+6x+9)+8-9}}}
{{{highlight(y=-1(x-3)^2-1)}}}------->{{{a(x-h)^2+k}}} where{{{system(h=x=3,k=y=highlight(1))}}}, highlighted: {{{a*k=(-1)(-1)=1}}}, oks!
Vertex: (3,1)
Axis of Symmetry: =3
Remember{{{system(a=-1,b=6,c=-8)}}}
Solving for Discriminant: {{{b^2-4ac=6^2-4*(-1)(-8)=36-32=4}}}
Then,
{{{x=(-b+-highlight(sqrt(b^2-4ac)))/(2a)=(-6+-sqrt(4))/(2*-1)}}}
{{{x=(-6+-2)/-2}}}
{{{x=(-6+2)/-2=-4/-2=2}}}
{{{x=(-6-2)/-2=-8/-2=4}}}
X-Intercepts{{{system(x=2,x=4)}}}
Y-Intercept: {{{f(x)=0}}}
{{{y=-(0)^2+6(0)-8}}}
{{{y=-8}}}
We'll see the graph,
{{{drawing(400,400,-6,6,-10,6,grid(1),graph(400,400,-6,6,-10,6,-x^2+6x-8),circle(2,0,.12),circle(4,0,.12),green(line(3,6,3,-10)), blue(circle(0,-8,.12)),circle(3,1,.12))}}}
c){{{y=-2x^2-12x-7}}}----->{{{y=-2(x^2+6x+7/2)}}}
Completing the square:
{{{y=-2(x^2+6x+9)+(7/2)-9}}}
{{{y=-2(x+3)^2+(7-18)/2}}}
{{{highlight(y=-2(x+3)^2-(11/2))}}}-->it follows {{{a(x-h)^2+k}}} where{{{system(h=x=-3,k=y=11)}}}
Vertex: (-3,11)------->Again, {{{y=a*k=-2*(-11/2)=11}}}
Axis 0f Symmetry= -3
For X-Intercepts:
Remember{{{system(a=1,b=6,c=7/2)}}}
WE 1st solve for the discriminant: {{{b^2-4ac=6^2-4*1*(7/2)=36-(28/2)=36-14=22}}}
Then, {{{x=(-b+-highlight(sqrt(b^2-4ac)))/(2a)=(-6+-sqrt(22))/(2*1)}}}
{{{x=(-6+-4.69)/2}}}
{{{x=(-6+4.69)/2=-1.31/2=highlight(-0.655)}}}
{{{x=(-6-4.69)/2=-10.69/2=highlight(-5.345)}}}
Y-Intercept: {{{f(x)=0}}}
{{{y=-2(0)^2-12(0)-7}}}
{{{y=-7}}}
We see the graph:
{{{drawing(400,400,-8,6,-8,12,grid(1),graph(400,400,-8,6,-8,12,-2x^2-12x-7),circle(-0.655,0,.15),circle(-5.345,0,.15),green(line(-3,12,-3,-8)),circle(0,-7,.12),circle(-3,11,.12))}}}
Thank you,
Jojo</pre>