Given the function f(x) = -2 |x-5/2|+6 and the graph g(x) shown below, over what interval do the functions f(x) and g(x) have the same minimum value?
(Since I cannot graph g(x), its equation is 2|x+0.5| -3 )
@amistre64 @whpalmer4

- anonymous

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- amistre64

what does it mean to have the same minimum value?

- anonymous

No idea. xD

- amistre64

well, it can mean 2 things to me ... what are your thoughts?

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## More answers

- anonymous

I'm assuming it means the smallest value where they intersect

- amistre64

that was one idea, does f(c) = g(c) ?? this doesnt sit right to me tho.
given an interval x = [a,b]
f and g have a minimum value within it such that say f(c) = g(d)

- amistre64

|dw:1443371069792:dw|

- amistre64

if we have this sort of interval, then we see we have 2 different minimums
|dw:1443371139444:dw|

- amistre64

what are our vertexes?

- amistre64

and do we have options since there are about an infinite number of intervals we can make that satisfy the results...

- anonymous

Im lost xd

- amistre64

what is our vertex of g?

- anonymous

The answer choices are
A [0,8]
B [1,7]
C [-2,5]
D [-5,1]
E None of these

- amistre64

|dw:1443371398297:dw|
oh good, choices, then we have something we can compare

- amistre64

what is our vertex of g, our biggest interval will then be along the y= axis of that vertex

- anonymous

-.5,3?

- amistre64

2|x+0.5| -3 )
-0.5, -3 is what it looks like to me, so when does f=-3?

- anonymous

Not sure

- amistre64

you have to solve for it ...
-2 |x-5/2|+6 = -3
-2 |x-5/2| = -3-6
|x-5/2| = (3+6)/2
etc ...

- anonymous

Ok so what would that be?

- amistre64

im not going to do the algebra for you ...

- anonymous

xd ok bare with my laziness gimme a sec

- anonymous

x = -2 and x = 7?

- amistre64

good, so this gives us the largest interval that the minimum of f and g are the same.
any option of x < -2 is bad, and any option of x>7 is bad ... can we rule any out?

- amistre64

outside of -2 to 7, the min of f drops below the min of g .... g cant go lower than its vertex by default

- anonymous

I can rule D out

- amistre64

D is bad, and A goes to 8 so it is bad for us too

- anonymous

So is the answer none of these?

- amistre64

|dw:1443372355236:dw|

- amistre64

now, if our interval does NOT include -0.5, then it cannot include -2 or 7
but our only options left have -2 or 7 involved. which one does not contain x=-0.5?

- anonymous

So it's none of these?

- amistre64

you havent answered my question ...

- anonymous

All of them?

- amistre64

if your not understanding what I am asking for, then say so ...

- anonymous

Yeah what are you trying to ask?

- amistre64

im asking for an interval which contains x=-0.5
draw me a number line, -2 to 7 .. and tic off -0.5 for me

- anonymous

|dw:1443372753726:dw|

- amistre64

now show me on that drawing, where 1 to 7 fits in ..

- anonymous

|dw:1443372851792:dw|

- amistre64

|dw:1443372874804:dw|
is -0.5 in the interval from 1 to 7?

- anonymous

n0pe.

- amistre64

is -0.5 in the interval from -2 to 5?

- anonymous

yes

- amistre64

then we should pick that one.

- anonymous

So the answer is C?

- amistre64

i would go with C yes

- anonymous

Gracias!

- amistre64

de nada

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