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In Exercises 45–48, let f(x) = (x - 2)2 + 1. Match the
function with its graph

In Exercises 4548 Let Fx X 22 1 Match The Function With Its Graph class=

Sagot :

whitneytr12

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!

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