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A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit. What is the value of $a+b+2c$? [asy] size(150); real ticklen=3; real tickspace=2; real ticklength=0.1cm; real axisarrowsize=0.14cm; pen axispen=black+1.3bp; real vectorarrowsize=0.2cm; real tickdown=-0.5; real tickdownlength=-0.15inch; real tickdownbase=0.3; real wholetickdown=tickdown; void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) { import graph; real i; if(complexplane) { label("$\textnormal{Re}$",(xright,0),SE); label("$\textnormal{Im}$",(0,ytop),NW); } else { label("$x$",(xright+0.4,-0.5)); label("$y$",(-0.5,ytop+0.2)); } ylimits(ybottom,ytop); xlimits( xleft, xright); real[] TicksArrx,TicksArry; for(i=xleft+xstep; i<xright; i+=xstep) { if(abs(i) >0.1) { TicksArrx.push(i); } } for(i=ybottom+ystep; i<ytop; i+=ystep) { if(abs(i) >0.1) { TicksArry.push(i); } } if(usegrid) { xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true); yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows); } if(useticks) { xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize)); yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize)); } else { xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize)); yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize)); } }; rr_cartesian_axes(-4,3,-2,9); real f(real x) {return 8-(x+1)^2;} draw(graph(f,-3.9,2.16,operator ..), red); [/asy]

Guest Nov 25, 2014

Best Answer 

 #1
avatar+92699 
+10

"c" is easy.... it's = 7

The vertex is at (h, k) = (-1, 8)   and the point    (2. -1) is on the graph

So we have

y = a(x - h)^2 + k

-1 =a(2- (-1))^2 + 8

-1 = a(3)^2 + 8      subtract 8 from both sides

-9 = a(9)     divide both sides by 9

-1 = a  

And the x coordinate of the vertex is given by  -b/2a

So we have 

-1 = -b/2(-1)  →  -1 = -b/-2 →  b = - 2

So.....our function is

y = -1x^2 - 2x + 7

So

a + b + 2c =

-1 + (-2) + 2(7) =

11

 

 

CPhill  Nov 25, 2014
 #1
avatar+92699 
+10
Best Answer

"c" is easy.... it's = 7

The vertex is at (h, k) = (-1, 8)   and the point    (2. -1) is on the graph

So we have

y = a(x - h)^2 + k

-1 =a(2- (-1))^2 + 8

-1 = a(3)^2 + 8      subtract 8 from both sides

-9 = a(9)     divide both sides by 9

-1 = a  

And the x coordinate of the vertex is given by  -b/2a

So we have 

-1 = -b/2(-1)  →  -1 = -b/-2 →  b = - 2

So.....our function is

y = -1x^2 - 2x + 7

So

a + b + 2c =

-1 + (-2) + 2(7) =

11

 

 

CPhill  Nov 25, 2014
 #2
avatar+92699 
0

Thanks for those points,  Melody....!!!!

 

CPhill  Nov 25, 2014
 #3
avatar+94106 
+5

This is a really good one for high school students to look at.  (Maybe years 9 to 11)

Chris has given you a nice solution. Thanks Chris.

Melody  Nov 25, 2014

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