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The graph of $y = f(x)$ is shown below.


For each point $(a,b)$ that is on the graph of $y = f(x),$ the point $\left( 3a - 1, \frac{b}{2} \right)$ is plotted, forming the graph of another function $y = g(x).$ As an example, the point $(0,2)$ lies on the graph of $y = f(x),$ so the point $(3 \cdot 0 - 1, 2/2) = (-1,1)$ lies on the graph of $y = g(x).$
[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 if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i 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(-5,5,-5,6);
draw((-4,4)--(-1,0)--(0,2)--(4,-4),red);
label("$y = f(x)$", (3,3), UnFill);
[/asy]

(a) Plot the graph of $y = g(x).$ Include the diagram as part of your solution.

(b) Express $g(x)$ in terms of $f(x).$

(c) Describe the transformations that can be applied to the graph of $y = f(x)$ to obtain the graph of $y = g(x).$ For example, one transformation could be to stretch the graph vertically by a factor of $4.$
 

 Jun 27, 2021
 #1
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0

(a) See the graph below.

 

(b) g(x) = 1/3*f(2x + 2).

 

(c) We stretch the graph horiztonally by a factor of 2, then stretch the graph vertically by a factor of 3, then shift down 2 units.

 

 Jun 27, 2021
 #2
avatar
0

Explaination for b)?

 Jun 27, 2021

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