Answer:
45) The function corresponds to graph A
46) The function corresponds to graph C
47) The function corresponds to graph B
48) The function corresponds to graph D
Step-by-step explanation:
We know that the function f(x) is:
[tex]f(x)=(x-2)^{2}+1[/tex]
45)
The function g(x) is given by:
[tex]g(x)=f(x-1)[/tex]
using f(x) we can find f(x-1)
[tex]g(x)=((x-1)-2)^{2}+1=(x-3)^{2}+1[/tex]
If we take the derivative and equal to zero we will find the minimum value of the parabolla (x,y) and then find the correct graph.
[tex]g(x)'=2(x-3)[/tex]
[tex]2(x-3)=0[/tex]
[tex]x=3[/tex]
Puting it on g(x) we will get y value.
[tex]y=g(3)=(3-3)^{2}+1[/tex]
[tex]y=g(3)=1[/tex]
Then, the minimum point of this function is (3,1) and it corresponds to (A)
46)
Let's use the same method here.
[tex]g(x)=f(x+2)[/tex]
[tex]g(x)=((x+2)-2)^{2}+1[/tex]
[tex]g(x)=(x)^{2}+1[/tex]
Let's find the first derivative and equal to zero to find x and y minimum value.
[tex]g'(x)=2x[/tex]
[tex]0=2x[/tex]
[tex]x=0[/tex]
Evaluatinf g(x) at this value of x we have:
[tex]g(0)=(x)^{2}+1[/tex]
[tex]g(0)=1[/tex]
Then, the minimum point of this function is (0,1) and it corresponds to (C)
47)
Let's use the same method here.
[tex]g(x)=f(x)+2[/tex]
[tex]g(x)=(x-2)^{2}+1+2[/tex]
[tex]g(x)=(x-2)^{2}+3[/tex]
Let's find the first derivative and equal to zero to find x and y minimum value.
[tex]g'(x)=2(x-2)[/tex]
[tex]0=2(x-2)[/tex]
[tex]x=2[/tex]
Evaluatinf g(x) at this value of x we have:
[tex]g(2)=(2-2)^{2}+3[/tex]
[tex]g(2)=3[/tex]
Then, the minimum point of this function is (2,3) and it corresponds to (B)
48)
Let's use the same method here.
[tex]g(x)=f(x)-3[/tex]
[tex]g(x)=(x-2)^{2}+1-3[/tex]
[tex]g(x)=(x-2)^{2}-2[/tex]
Let's find the first derivative and equal to zero to find x and y minimum value.
[tex]g'(x)=2(x-2)[/tex]
[tex]0=2(x-2)[/tex]
[tex]x=2[/tex]
Evaluatinf g(x) at this value of x we have:
[tex]g(2)=(2-2)^{2}-2[/tex]
[tex]g(2)=-2[/tex]
Then, the minimum point of this function is (2,-2) and it corresponds to (D)
I hope it helps you!
