Question

graph the function and state the vertex, axis of symmetry, and the intercepts, if any.

f(x)=x^2-6x+8

Answer 1

This is easy.

Answer 2

First you have to find the "zeroes" or the x-intercepts.

Answer 3

so do i replace all x's with 0?

Answer 4

No

Answer 5

You have to replace Y AS 0, because the x-intercept is what x is when y = 0.

Answer 6

So we have 0 = x^2 - 6x + 8

Answer 7

ok

Answer 8

Now we have to factor.

Answer 9

do you have to get rid of the ^2?

Answer 10

No

Answer 11

im sorry math is not my strong suit, this stuff really confuses me =(

Answer 12

Well, later on we do.

Answer 13

It's alright.

Answer 14

So we have to find out the squares of the first and last terms, do you know what they are?

Answer 15

Well we get x from the first term, and we use 8 as our last term.

Answer 16

So 8 * x = 8x so we need to find out what the factors of 8 are that add up to the MIDDLE TERM 6x.

Answer 17

Can you state some factors of 8?

Answer 18

1,2,4,8

Answer 19

Right, so let's try 1 and 8.

Answer 20

Remember, they both have to add up to 6 and multiply to 8.

Answer 21

1 x 8 = 8. But 1 + 8 = 9. We're looking for a 6, not a 9.

Answer 22

ok

Answer 23

2 and 4

Answer 24

Yes, you got it!

Answer 25

So we replace 6x with 4x and 2x.

Answer 26

0= x^2-(4x+2x)+8?

Answer 27

Uh-oh

Answer 28

?

Answer 29

I made a mistake, sorry. We need 2 things that add up to -6 and multiply to 8.

Answer 30

so -2 and 4

Answer 31

-2 and -4*

Answer 32

Yes

Answer 33

So we have 0= x^2 - 4x - 2x + 8?

Answer 34

Now we're going to separate the first two terms and the last two terms with a parenthesis:

Answer 35

0 - (x^2 - 4x) - (2x + 8)

Answer 36

ok

Answer 37

We have to find out the GCF or greatest common factor of each parenthesis, can you tell what the GCF of the first parenthesis is?

Answer 38

1?

Answer 39

No, look again. Remember it can be a variable.

Answer 40

x

Answer 41

Yes.

Answer 42

So we're going to work backwards with the first parenthesis:

x(x^2 - 4x)
We are DIVIDING x by each term, since we're working backwards

Answer 43

ok

Answer 44

So basically we're removing 1 x from all the parenthesis:
x(x^2 - 4x)
Removing 1 x from x^2 would give us a regular x.
x(x - 4x)
And removing x from -4x will give us a -4.
x(x - 4)

So we finished our first parenthesis.

Answer 45

Can you tell me the GCF of the 2nd parenthesis?

Answer 46

2

Answer 47

so it would be 2(2x+4)?

Answer 48

Yes!

Answer 49

2(x+4)*

Answer 50

You made a mistake, remember that your 2x is a NEGATIVE 2x.

Answer 51

so 2(-x+4)?

Answer 52

No, your GCF will be a -2.

Answer 53

You want to keep x positive.

Answer 54

-2(x+4)

Answer 55

Yes, almost perfect. Remember 8 / -2 = -4, not 4.

Answer 56

-2(x-4)

Answer 57

Yes

Answer 58

So we have 0 = x(x - 4) - 2(x - 4)
We can combine x - 4, because they're the same(If these terms aren't the same, then you know you made a mistake). And we have to combine the terms outside of both parentheses.

So we have (x - 2)(x - 4)

Do you understand how I got (x - 2)?

Answer 59

no =(

Answer 60

oooh wait! the x and -2 outside of the parentheses!

Answer 61

Yes

Answer 62

You need to combine those.

Answer 63

ok

Answer 64

So now that we figured out that form we can finish the problem.

0 = (x - 2)(x - 4)

Answer 65

We have to break each parenthesis down and add a '= 0' to it.

So for the first parenthesis:

0 = x - 2
Now we just figure out what x is by adding 2 to both sides.
2 = x
x = 2

Can you figure out the 2nd parenthesis out?

Answer 66

x=4

Answer 67

Yes, so we have: x = 2 and x = 4. We know that the parabola crosses the x-axis at 2 and 4.

Answer 68

Those are your x-intercepts.

Answer 69

ok

Answer 70

The y-intercept is what y is when x = 0. So we can go back to the original equation and change all the x's to 0.

f(x) = (0)^2 - 6(0) + 8
f(x) = 0 - 0 + 8
f(x) = 0 + 8
f(x) = 8

So our y-intercept is 8.

Answer 71

would it be (0,8) for the y intercept and (2,0) (4,0) for x?

Answer 72

Yep.

Answer 73

ok =)

Answer 74

Now we have to find the vertex.

Answer 75

The formula for the vertex is -b/2a.

Answer 76

Your equation is in the form of a^2 - bx + c

So what is 'a' and 'b'?

Answer 77

x and -6

Answer 78

Not quite, the "a" has to be a number.

Answer 79

1

Answer 80

x by itself can also be what?

Answer 81

Yes, that's it.

Answer 82

So we have 1 and -6.

We can plug these into our equation:
-1/2(-6)
Can you figure it out?

Answer 83

My bad it's:

-6/2(-1)

Answer 84

6/-2

Answer 85

-3

Answer 86

Ugh, I made another mistake it's
6/2(1)
6/2
3

Answer 87

3 is the x-value of our vertex. We can plug it into our equation to get the y-intercept.

Answer 88

oh ok

Answer 89

f(x) = x^2 - 6x + 8
f(x) = (3)^2 - 6(3) + 8

Answer 90

f(x) = (3)^2 - 6(3) + 8
f(x) = 9 - 18 + 8
f(x) = -9 + 8
f(x) = -1

So the y-value of our vertex is -1.

Answer 91

yes i got -1

Answer 92

vertex is (3,-1) ?

Answer 93

So our vertex is: (3, -1). Our y-intercept is (0, 8), and our x-intercepts are (2, 0) and (4, 0).

Answer 94

Yes, it's (3, -1)

Answer 95

The axis of symmetry will be our x-value, or 3.

Answer 96

oh ok awesome!

Answer 97

That's everything, that's how you figure it out by looking at it from our equation. You can graph it here: https://www.desmos.com/calculator

And you will get all the instant answers.

Answer 98

thank you so much!!! =)