Question

Will Give Medal!!!

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

x-intercepts are the zeros of the function, where the graph crosses the x-axis, aka at y=0.

Therefore set the equation equal to zero.
f(x) is another way of saying y.
\[y=–16x^2+24x+16\]
\[0 = –16x^2 + 24x + 16\]

Answer 2

Yes.. that i know how to do... I know also that you then have to find the GCF and then factor that out... when you do this, i get

8(-2x^2 +3x + 2) (i think this is right...?)

Then i know you are supposed to multiply -2 and 2 (you get 4) then find the factors of 4 and see which of the two factors you find that you can add together in order to find the middle term (3)

I THINK this is how you do it... am i right? If i am right i have no idea what i am doing wrong... if i am wrong i have no idea what to do...

Answer 3

Oh ok... never saw how to do that before lol but it makes since... what do i do after that?

Answer 4

Actually you can divide the whole equation by 8 since it's equal to 0. Divide both sides by 8. (0 divided by 8 is still 0). It becomes:
0 = -2x^2 + 3x + 2

(sorry divided wrong previously)

Answer 5

Oh ok thats what i got when i did the GCF.. so now i am confused on what to do next...

Answer 6

Now you can factor. You need to numbers that add to the coefficient of x (which is 3) and multiply to the product of the constant and coefficient of x^2 (product is -4).

Answer 7

ok... sorry im a little confused

Answer 8

This is how I like to organize it.

Answer 9

Do you understand how I got the 3 and 4?

Answer 10

I think that 3 is the middle term and you got 4 by multiply 2 times 2...?

Answer 11

Yes.
Now the two numbers you found, 4 and -1, you put into the equation like this:
0 = -2x^2 + 3x + 2
0 = -2x^2 + 4x -x + 2

Notice the 3x is broken down into 4x and -1x, that's where the numbers come in; they're the coefficients now.

Answer 12

Oh ok... but why would you put 4 and 1 in place of 3?

Answer 13

Nevermind i see what you wrote... ok so now we solve the problem regularly?

Answer 14

If i did i would get

-2x(x+2) -1(x+2)... am i right?

Answer 15

Not completely..

Answer 16

ok...what did i do wrong

Answer 17

You have 0 = -2x^2 + 4x -x + 2

2x(x+2) -1(x+2)
If you distribute back 2x * x does not equal the first term -2x^2.
And -1 * 2 does not get 2

Answer 18

Oh i see... so what did we do wrong because thats how you would solve it... i think

Answer 19

It should have some negatives in there...
-2x^2 + 4x - x + 2

You chose to take out 2x.
2x(-x+2) - x + 2

And a rule of factoring further is for the second part to have (-x+2).
The only way is to factor out a positive 1.
2x(-x+2) +1(-x+2)

Answer 20

Oh... ok i see i shouldnt have put the negatives on the outside of the parenthesis... So after that normally to find the x's (like it says to do in part a) i would then take 2x and 1 and put them together in a parenthesis and then thake the (-x+2)

So i would have (2x+1)(-x+2)

But now i dont understand what to do next...

Answer 21

Yes and that equals 0. I forgot to include the 0 in the others previously.

0 = (2x+1)(-x+2)

So what makes the right side equal 0? Since it's multiplication, either one of them has to equal 0.
So,
0 = 2x + 1 and 0 = -x + 2

Answer 22

im confused...

Answer 23

my bad it should be or instead of and
0 = 2x + 1 or 0 = -x + 2

Answer 24

hm which part?

Answer 25

the part where you said

So what makes the right side equal 0? Since it's multiplication, either one of them has to equal 0.

Answer 26

That's the zero product of multiplication. 0 times anything is 0.
So since (2x+1)(-x+2) has to equal 0, either 2x+1 has to equal 0 or -x + 2 has to equal 0.
So in the end you get two solutions (two zeros) because of the "or" part

Answer 27

Oh yea thats right ok... so to make (2x+1) equal zero we would have to do

2x + 1 - 3 = 0 - 3

then the x intercept would be (-3,0)

I think...

Answer 28

You're just solving the equation
2x + 1 = 0

Subtract 1 from both sides, then divide by 2

Answer 29

If i did that i would get x = -0.5...

Answer 30

Yes that's correct. That's one of the solutions. You also have -x+2 = 0

Answer 31

2

Answer 32

x=2

Answer 33

Correct

Answer 34

YAY!!!!

Answer 35

What about this... if you have time

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.

Answer 36

Since there's a negative in front (-16) the graph will slope downward like an upside down v ( like a tepee). Therefore it will have a maximum since there's the tip at the top and the bottom (range) goes down to infinity and won't end.