Question

The function f(x) = −x2 + 28x − 192 models the hourly profit, in dollars, a shop makes for selling sodas, where x is the number of sodas sold and f(x) is the amount of profit.

Part A: Determine the vertex. What does this calculation mean in the context of the problem? (5 points)

Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)

Answer 1

this could help you graph the equation.. https://www.desmos.com/calculator

Answer 2

Ok, so I'm not good at math

Answer 3

so how would I graph it

Answer 4

i think the 2 after -x is supposed to be like lower than the number right? i don't think it lets u type that on here

Answer 5

the 2 after the x is supposed to be an exponent

Answer 6

ohhh

Answer 7

oh okay lemme graph that

Answer 8

\[f(x) = −x^2 + 28x − 192\]

Answer 9

okay so heres the graph so you know what I'm talking about https://www.desmos.com/calculator/4sg7vl6plb

Answer 10

do you know what a vertex is?

Answer 11

it's like the highest point in the graph isn't it?

Answer 12

well it can be the highest or the lowest depending on if its a minimum or maximum

Answer 13

in this case it is the highest which means its a maximum

Answer 14

Okay

Answer 15

so the vertex would be (14,4)...

Answer 16

where do you get (14, 4) from?

Answer 17

one sec ill show you

Answer 18

because on the graph it's showing this

Answer 19

the highest point is the vertex

Answer 20

if it opened upwards then the lowest point would be the vertex

Answer 21

I was looking at the wrong spot my bad

Answer 22

its okay dw

Answer 23

Would you mind if I explained how to get the answer to part (sorry to disturb, I just wanna help) @xhikari

Answer 24

I was looking where the arch went past the X axis on both sides

Answer 25

part A

Answer 26

*

Answer 27

when it goes past the x axis those are the 'zeros' aka x intercepts which is part b

Answer 28

Oh okay, so theres 2 x intercepts ?

Answer 29

yes

Answer 30

ima go get somethin to drink rq if you don't mind @_@

Answer 31

yea np

Answer 32

it's like 80 degrees outside and it's almost 7 in the afternoon @_@

Answer 33

90 degrees sorry

Answer 34

anyways so, theres 2 x intercepts per graph, but what happens to 192 in the equation?

Answer 35

i dont think we need that bc its just asking for the vertex, x-intercepts, and what those mean.

Answer 36

Oh okay, so 192 doesn't count towards Part A nor B?

Answer 37

i dont think so. so we know that the vertex is (14,4) and the x-intercepts are 12 and 16

Answer 38

im gonna use a new method to solve this problem instead, are u familiar with completing the square?

Answer 39

sorry my mum called me down

Answer 40

do you still need me orr??

Answer 41

yeah, and it's alr

Answer 42

okay so i told you what the vertex means (maximum number of hourly profit (someone correct if I'm wrong)) so you have part A and now for part b we need to determine what the x interceots represent

Answer 43

So part A is maximum number of hourly profit?

Answer 44

yes but also include the vertex itself which is (14,4)

Answer 45

ok, so lets use the formula \[x = -\frac{ b }{ 2a }\]

In this case

a = -1
b = 28

When we plug those numbers in we get...\[-\frac{ 28 }{ 2(-1) }\]


What do you get when you solve this equation @angelcook2007

Answer 46

sorry I took so long :(

Answer 47

oh okay, so like "The vertex is (14, 4), and this shows the maximum number of hourly profit"?

Answer 48

sooo... what do you get when you solve the equation?

Answer 49

sorry, my questioncove tab decided to die

Answer 50

anyways

Answer 51

solve \[\frac{ -28 }{ 2 -1}\]

Answer 52

???

Answer 53

yea... basically this... try solving it :)

Answer 54

what did you get?

Answer 55

14?

Answer 56

That's correct, now you find your x-vaule for your vertex :D

(14, )

Now to find the y-value, you just substitute the `x` in the problem with 14 then solve for `y`

\[f(x) = −x^2 + 28x − 192\]
\[y = −(14)^2 + 28(14) − 192\]

Can you try `solving for y`?

Answer 57

2?

Answer 58

not 2

Answer 59

4?

Answer 60

idek where 2 came from lol

Answer 61

correct, the y-value would be 4. That means the vertex is `(14,4)`

Answer 62

does that make sense ?

Answer 63

me and @xhikari found the vertex

Answer 64

bro it's so hot outside lol

Answer 65

yea, but you have to explain how you got the answer, not just say that u went on desmos ;/

Answer 66

oh ok lol

Answer 67

And you have to use the equation to find the answer lol

Answer 68

i'm honestly tempted to go eat some ice ngl

Answer 69

how hot is it where you live lol

Answer 70

it's almost 90 at my house

Answer 71

very hot, but lets continue and not get off track lol, do you still need help with part b?

Answer 72

lol

Answer 73

The x intercepts are 12 and 16 but idk what they represent

Answer 74

Well, the x-intercepts basically means that the shop will make 0 dollars when they sell 12 and 16 sodas

soooo sorry for taking so long, im studying for my finals tomorrow lol

Answer 75

Do you need help on finding the x-intercepts btw ?

Answer 76

it's alright

Answer 77

oh lol

Answer 78

I randomly got a medal btw, what does that mean?

Answer 79

Oh

Answer 80

okay, thanks lol

Answer 81

so the x intercepts mean that when they sell 12 and 16 sodas, they make a $0 profit or somthing?

Answer 82

oh okay, thank you