Question 184808
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Given: {{{y-2=-1/2(x-4)^2}}}--->{{{y-2=(-1/2)(x^2-8x+16)}}}, distribute {{{-1/2}}} and transfer {{{-2}}} to the right.
{{{y=(-1/2)x^2-(-8/2)x+(-16/2)+2=(-1/2)x^2+4x-8+2}}}
{{{y=(-1/2)x^2+4x-6}}}, EQN 1
First, we'll find the vertex thru Vertex form: {{{y=a(x-h)^2+k^2}}}
where{{{system(h=x,k=y)}}} of the vertex
we complete the square:
Divide the whole eqn by {{{-1/2}}}:
{{{y/(-1/2)=((-1/2)x^2+4x-6)/(-1/2)}}}--->{{{-2y=(cross(-1/2)x^2/cross(-1/2))+(4x/(-1/2))-(6/(-1/2))}}}
{{{-2y=x^2-8x+12}}}
{{{-2y=(x^2-8x+highlight(16))+12-highlight(16)}}}
{{{-2y=(x-4)^2-4}}}
{{{cross(-2)y/cross(-2)=((x-4)^2-4)/-2}}}
{{{y=(-1/2)(x-4)^2-(4/-2)}}}
{{{y=(-1/2)(x-4)^2+2}}}
You see, {{{highlight(h=x=4)}}} & {{{highlight(k=y=2)}}}--->VERTEX (4,2)
Next:
Let {{{Fx=0}}}, as per EQN 1
{{{y=(-1/2)(0)^2+4(0)-6}}}
{{{highlight(y=-6)}}}, Y Intercept
Next:
For the X Intercepts, we solve the Eqn1:
by QUADRATIC, where{{{system(a=-1/2,b=4,c=-6)}}}
Solving for he discriminant: {{{b^2-4ac=4^2-4(-1/2)(-6)=16-12=4}}}
Therefore,
{{{x=(-4+-sqrt(4))/(2*-1/2)=(-4+-2)/-1}}}
{{{x=(-4+2)/-1=-2/-1=highlight(2)}}}
Also, {{{x=(-4-2)/-1=-6/-1=highlight(6)}}}
Then we see the graph, plotting all intercepts:
{{{drawing(400,400,-8,8,-8,8,grid(1),graph(400,400,-8,8,-8,8,(-1/2)x^2+4x-6),blue(circle(4,2,.12)),green(circle(2,0,.12)),green(circle(6,0,.12)),circle(0,-6,.12))}}}
Thank you,
Jojo</pre>