Question

Use the function f(x) = x2 − 2x + 8 and the graph of g(x) to determine the difference between the maximum value of g(x) and the minimum value of f(x).

a parabola that opens down and passes through 0 comma 3, 3 comma 12, and 5 comma 8


2

5

7

12

Answer 1

The difference between the maximum value of g(x) and the minimum value of f(x) is:

5

Explanation:

• The function f(x) is given by:

`f(x)=x^2-2x+8`

which in vertex form is given by:

`f(x)=(x-1)^2+7`

Now, we know that:

`(x-1)^2\geq 0\\\\(x-1)^2+7\geq 0+7\\\\(x-1)^2+7\geq 7\\\\i.e.\\\\f(x)\geq 7`

Hence, the minimum value of the function f(x) is: 7

• Also, the function g(x) is a parabola that opens down and passes through (0,3) , (3,12) and (5,8) .

Let the equation for g(x) be:

`g(x)=ax^2+bx+c`

Now with the help of the passing through three points we get:

when it passes through (0,3)

`3=a* 0+b* 0+c\\\\i.e.\\\\c=3`

when it pass through (3,12)

i.e.

`9a+3b+c=12\\\\i.e.\\\\9a+3b+3=12\\\\9a+3b=12-3\\\\9a+3b=9\\\\3a+b=3---------(2)`

Also it pass through (5,8)

i.e.

`25a+5b+c=8\\\\25a+5b+3=8\\\\25a+5b=8-3\\\\25a+5b=5\\\\5a+b=1-------(2)`

On subtracting equation (2) from equation (1) we have:

`a=-1`

and by putting the value of a in equation (1) we have:

`b=6`

Hence, we get:

`g(x)=-x^2+6x+3\\\\g(x)=-(x-3)^2+12`

Now, we know that:

`(x-3)^2\geq 0\\\\-(x-3)^2\leq 0\\\\-(x-3)^2+12\leq 12`

This means that the maximum value of g(x) is: 12

The difference between the maximum value of g(x) and the minimum value of f(x) is:

12-7=5

Answer 2

Answer: b) 5

Explanation:

As given by the graph, the maximum value of g(x) is 12

Now, let's find the Axis Of Symmetry of f(x). Then plug in the x-value into f(x) to find the minimum value of f(x).

`x=(-b)/(2a)=(-(-2))/(2(1))=(2)/(2)=1\\\\f(x)=x^2-2x+8\\f(1) = 1^2-2(1)+8\\.\qquad =1-2+8\\.\qquad=7`

Difference = max of g(x) - min of f(x)

= 12 - 7

= 5