Question

Someone help please.

An expression is shown below:
f(x) = -16x2 + 24x + 16

Part A: What are the x-intercepts of the graph of f(x)? Show your work.

Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work.

Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph.

Answer 1

@triciaal @zepdrix @mathmale @mathstudent55

Answer 2

@Directrix @Hero @ParthKohli @pooja195 @Mehek14 @mayankdevnani @Jaynator495

Answer 3

Sup :d

Answer 4

Sup

Answer 5

sup sup sup sup sup sup sup

Answer 6

The function crosses the x-axis when.... the y-coordinate is zero.
Hopefully that makes sense.

So to find our x-intercepts, where the function crosses the x-axis,
we'll replace our y with zero.
And then solve for x.

Answer 7

\[\large\rm f(x)=-16x^2+24x+16\]\[\large\rm ~~~~0=-16x^2+24x+16\]

Answer 8

The next few steps should feel familiar,
it's what you've been doing a lot of, factoring :)

Answer 9

@zepdrix I gotta go, be back in 5 minutes. sorry lol

Answer 10

@zepdrix Dude, really sorry, but I have to leave for like 6 hours Sorry.

Answer 11

k c:

Answer 12

@zepdrix I'm back lol

Answer 13

@zepdrix So I need your help now. I get that y has to be 0 and then solve for x, but what then?

Answer 14

@zepdrix All right, ill be back in 15 - 20 minutes

Answer 15

then what? then do it, solve for x :o
get your factors.

Answer 16

Alright, I'll try

Answer 17

@needhelpstudying: Please, would you plan your time so that you have enough time in which to complete a whole problem before excusing yourself? Constantly excusing yourself could be interpreted to be disrespect for the person helping you.

zepdrix is correct in asking you to solve the equation \[-16x^2+24x+16 = 0.\]

Answer 18

What is the largest number (greatest common factor) that will divide into 16 and 24 with no remainder? Factor out the GCF. Rewrite the equation in this factored form.

Which methods for solving a quadratic equation do you know: For starters: completing the square, factoring, graphing, quadratic formula.

Answer 19

@mathmale Sorry, some family friends came to my house, my parents didn't really give me any warning at all lol.

Should I be factoring this by grouping?

−16x2+24x+16

Answer 20

@zepdrix If ur online, I can start now

Answer 21

as mentioned above first always look for GCF ( greatest common factor )
is there any thing common ? if yes take it out first and then factor it

Answer 22

eyy broski.
grouping? naw not just yet. do that ^

Answer 23

They're all even numbers, so they at least have a 2 in them.
Anything larger as a factor in each?

Answer 24

@zepdrix Sorry, laptop shut off. Alright.

GCF Of 8 and 24 is 8. so GCF would be 8x.

Factor it out:

-16x2 + 24x + 16

8x(-2x + 3) + 16

Correct?

Answer 25

No no no, we're looking to factor something out of EVERY TERM to start this one out.

Answer 26

The last term, 16, has no x, so we can't pull 8x out.
We'll have to result to pulling just 8 out of everything, ya?

Answer 27

resolve*

Answer 28

\[\large\rm -16x^2+24x+16\]\[\large\rm 8(-2x^2+3x+2)\]Understand that step?

Answer 29

@zepdrix Yeah, I get it now. I kinda ignored that last term, sorry

Answer 30

Yeah, so i get that step

Answer 31

These usually work out a little nicer when the squared term is POSITIVE.
So let's apply another step, hope this one doesn't confuse you too much.
We're going to factor -1 out of each term.
Essentially what that does is, it just changes the `sign` of everything.\[\large\rm -8(2x^2-3x-2)\]

Answer 32

HOORAY FOR CHANGING SIGNS!

Answer 33

Wait, so what do you do to the -8 that's already outside the parantheses?

Answer 34

Well, recall that we were letting this entire thing equal 0.\[\large\rm 0=-8(2x^2-3x-2)\]So we can do something clever here.
We can divide both sides by -8,
our equation becomes this:\[\large\rm 0=2x^2-3x-2\]

Answer 35

NEVER NEVER NEVER NEVER NEVER

BTW It's -2x2

Answer 36

@zepdrix Hello?????????????????????

Answer 37

whut?

Answer 38

The -2x^2 became 2x^2 when we took the negative out of everything.

Answer 39

Wait

You wrote

0=2x2−3x−2

Should it be

0 / 8=2x2−3x−2

Answer 40

Yes :)
And what is 0 divided by -8?

Answer 41

@Needhelpstudying is amazing?

Answer 42

lololololololol 0 - 8 is ZEPDRIX!

Answer 43

take it serious.

Answer 44

0 / -8 = 0. Then the shark came and bit off my head. The End.

Answer 45

@zepdrix

Answer 46

@ganeshie8 Can you help?

Answer 47

which part ?

Answer 48

Pretty much everything

Answer 49

\[\large\rm 0=2x^2-3x-2\]

have you tried to factor this ?

Answer 50

I don't think anything can be factored from it.

Answer 51

pick two numbers such that
1) product is "-4"
2) sum is "-3"

Answer 52

GAAH How do you do those problems again?

Answer 53

@ganeshie8 Is there any formula for finding the answer, or is it just guess and check?

Answer 54

Factors of -4

-4, 1

-1, 4

2, -2

Answer 55

The answer is -4, 1. Correct @ganeshie8 ?

Answer 56

Yes, now rewrite the middle term -3x as -4x + 1x

Answer 57

\[0=2x^2−x−2\]

Answer 58

\[\large\rm 0=2x^2\color{red}{-3x}-2\]

is samea as

\[\large\rm 0=2x^2\color{red}{-4x+1x}-2\]

Answer 59

Next, group first two terms
group last two terms

Answer 60

Thats strange, when I wrote the equation it didn't include everything I typed....

Answer 61

0=(2x2−4x) + (1x−2)

Answer 62

happens... so what is the common factor in first two terms ?

Answer 63

2x?

Answer 64

and 1 for the second term?

Answer 65

Yes, pull that out

Answer 66

alright

0=2x(x − 2) + 1(x − 2)

Answer 67

0 = (2x + 1)(x − 2)

Answer 68

Correct?

Answer 69

Looks good. For that product to be really equal to 0, we must necessarily have either one of the factors equal to 0 :

0 = 2x + 1

or

0 = x - 2

Answer 70

you can solve x

Answer 71

x = -1/2

x = 2

?

Answer 72

Yep! those are the x intercepts.

The graph of given function f(x) cuts the x axis at -1/2 and 2

Answer 73

WOOHOOO!!!

Answer 74

I just realized after all this i've only solved part a. :(

Answer 75

Lets graph f(x) quick and verify part a

Answer 76

K, desmos?

Answer 77

Yeah, it's correct. I just checked.

Answer 78

good

Answer 79

On to part b

Answer 80

For part b :

The leading coefficient of f(x) is -16 which is negative. Therefore the vertex of f(x) is going to be a minimum.

Answer 81

wait wait wait, what is a vertex?

Answer 82

Look at the graph again. How does it look ? Do you see any "hill top" ?

Answer 83

In our case, the point on the top of the hill is called the vertex point.