Question

Consider the two functions.

f(x) = x² – 8x + 7


Do the minima of the two functions have the same x-value? yes or no


Which of the functions has the greater minimum? g(x) or f(x)

(second function is the graph below)

Answer 1

`f(x)` has the greater minimum at
`(4,-9)`

Explanation:

The given function is

`f(x)=x^(2)-8x+7`

Additionally, the graph attached shows the function
`g(x)` which as a minimum at (4,-4).

So, let's find the minimum of
`f(x)`, which has coordinates
`(h,k)` where
`h` is defined as

`h=-(b)/(2a)`

Where
`a=1` and
`b=-8`, replacing these values, we have

`h=-(b)/(2a)=-(-8)/(2(1))=4`

So, the other coordinate is defined as
`k=f(h)`, so replacing the value in the function, we have

`f(x)=x^(2)-8x+7\\f(4)=(4)^(2)-8(4)+7=16-32+7=-9`

So, the minimum of
`f(x)` is at
`(4,-9)`.

Therefore,
`f(x)` has the greater minimum at
`(4,-9)`

Answer 2

They have the same x-value

f(x) has the greater minimum

Explanation:

To find the vertex of a second degree equation, in this case the minimum value, we can use the following equation:

x = -b / 2a

Remember that a second degree equation has the following form:

ax^2 + bx + c

so a = 1, b = -8 and c = 7. Now you have to substitute in the previous equation

x = - (-8) / 2(1)

x = 8 / 2

x = 4

This means that the two functions have the same x-value.

The y value of f(x) would be

f(4) = (4)^2 - 8(4) + 7

f(4) = 16 - 32 + 7

f(4) = -9

So the vertex, or minimun value of f(x) would be at the point (4, -9).

The vertex, or minimun value of g(x) is at the point (4, -4).

So f(x) has a minimum value of -9 and g(x) a minimum value of -4.