The graph of function f is shown on the coordinate plane. Graph the line representing function g, if g is defined as shown below. G(x)=1/2f(x+2)

Accepted Solution

Answer:[tex]g(x)=x-1[/tex]Step-by-step explanation:Any problem of this kind should first be approached by solving the initial function. This question is based on curve translation on x-axis. But, there can be questions asked on translation in both axes, rotation of curve about origin, or both combined.  Initial function f(x) is a straight line from picture.Common equation of straight line is[tex]y=mx+c[/tex]m= slope of the linec= y-interceptSlope of the line:[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1}}[/tex]Let those points on graph be A(3,0) and B(0,-6)Therefore, slope of line AB, using formula is:[tex]m=\frac{0-(-6)}{3-0}[/tex]m=2And c=(-6)And equation of line AB is: [tex]y=2x-6[/tex][tex]f(x)=2x-6[/tex]Now this line is translated backwards by 2 units, (x) becomes (x+2)Let us consider a X, such that, [tex]X=x+2[/tex][tex]f(X)=2X-6[/tex][tex]f(x+2)=2(x+2)-6[/tex][tex]f(x+2)=2x+4-6[/tex][tex]f(x+2)=2x-2[/tex]Now, there is another function such that, [tex]g(x)= \frac{1}{2} f(x+2)[/tex][tex]g(x)= \frac{1}{2}(2x-2)[/tex][tex]g(x)= x-1[/tex] ; according to your question.In the picture, they are asking for a function, [tex]g(x)= \frac{-1}{2} f(x+2)[/tex]Then, [tex]g(x)= -x+1[/tex] ; according to the picture.Here, I have attached a file showing all the graphs for clear understanding.