Question

write polynomial functions with these features.
a) a linear function whose graph has x-intercepts 4.
Please help!!

Answer 1

x-intercepts? plural?

Answer 2

Only 1 x-intercept?

Answer 3

sorry its singular
x intercept of 4

Answer 4

x=4

Answer 5

And what degree polynomial are you instructed to write?

Answer 6

it says linear.

Answer 7

a linear function i guess

Answer 8

But x=4, therefore (x-4) is one linear factor

Answer 9

That would be degree 1

Answer 10

So according to the given information, you can make up another one

Answer 11

You cannot have a linear function with more than one x-intercept.

Answer 12

linear is y=mx+b right?

Answer 13

yes

Answer 14

so would it be y=4x+0?

Answer 15

So x-4=0

Answer 16

Yes, but what would be the function? y = ?

Answer 17

y=x-4

Answer 18

yes

Answer 19

Or f(x)=x-4

Answer 20

oh so where does the b go?

Answer 21

@pretyroze

Do you understand that?

Answer 22

b is the constant 4

Answer 23

i thought b was x?

Answer 24

@pretyroze b = 0, so y = 4x + 0 which simplifies to y = 4x.

Answer 25

No x is x, and m is the number before x, so in 2x-3 for example, 2 is m, x is x, and -3 is b

Answer 26

@pretyroze b is a contant, it represents the y-intercept of the straight line.

Answer 27

that makes a lot of sense

Answer 28

what if the question asks for a quadratic function whose graph has only one x intercept of 4? how would i set that up?

Answer 29

@pretyroze ...sorry, the equation is y = x -4...that has an x-intercept of x = 4.

so in that equation, -4 is the y-intercept.

Answer 30

why do you subtract 4?

Answer 31

oh i see

Answer 32

Look at the grap of y = x -4. You can see that x = 4 is an x-intercept.

Answer 33

yes and the y is a negative 4

Answer 34

yes, y-intercept of - 4.

Answer 35

what if the question asks for a quadratic function whose graph has only one x intercept of 4? how would i set that up?

Answer 36

You would have to make up the second intercept because a quadratic always has 2

Answer 37

(x-4)(x+2) and then factor

Answer 38

so would it be the conjugate of 4 which is -4?

Answer 39

Or whatever number in substitute of 2

Answer 40

where did the 2 come from

Answer 41

Made it up

Answer 42

You can't have a definite quadratic with only one x-intercept

Answer 43

ohh i see

Answer 44

so i can plug in any number?

Answer 45

So if they said the zero (x-intercept) is 3, than you would have to make up the other

Answer 46

But if they said it is 1+3i, (a complex zero) than you would have to multiply it by it's conjugate, (1-3i)

Answer 47

But yes, you can plug in any number. As long as only 1 x-intercept is given

Answer 48

ohhh so conjugates are only used with complex numbers?

Answer 49

Exactly

Answer 50

Ex. 1+5i. 1-5i 3+2i, 3-2i

Answer 51

And they are factored like this (x-(1+3i))(x-(1-3i))

Answer 52

Conjugates are also with irrational numbers, 2 + sqrt(3) and 2 - sqrt(3).

Answer 53

^

Answer 54

Not just for comples numbers.

Answer 55

Forgot about that ^

Answer 56

Also to remember, every degree up the polynomial goes, the number of zeros does as well.
Linear: 1
Quadratic: 2
3rd degree: 3
4th degree: you get the point

Answer 57

what if it asks us to do the same thing but for a cubic functon whose graph has only one x intercept of 4. I know that a cubic function is y=a(x-r1)(x-r2)(x-r3)

Answer 58

Then you have to make up 2 zeros

Answer 59

You probably won't get a question like that though

Answer 60

A cubic function means 3rd degree polynomial, therefore it has 3 zeros (x-intercepts)

Answer 61

lol its on my homework

Answer 62

Really?

Answer 63

yeaa and my teacher isnt the best

Answer 64

@evan97 @pretyroze

A cubic polynomial has 3 roots, but not necessarily 3 x-intercepts. Example...the roots may be x = 3, 2+3i, and 2 - 3i.

You can't plt the two imginary numbers on the real axis, so that polynomial will have one x-intercept.

Answer 65

In polynomials, you have to be careful not to equate roots with x-intercepts.

Answer 66

Yes but you can't factor a cubic function given only 1 real root correct?

Answer 67

but not all cubic polynomials have imaginary numbers correct?

Answer 68

Yes, you can.

Answer 69

Correct, the 3 roots may be either all 3 real numbers or 1 real number and 2 imaginary numbers. No other combination is possible for a cubic polynomial.

Answer 70

so in my case what would it be with an x intercept of 4

Answer 71

Yeah I'm wondering the same thing

Answer 72

@pretyroze @evan97

As an exmaple, the equation x^3 - 10x^2 + 37x - 57 = 0 has one real root and two imaginary roots. And this polynomial is factorable into (x-4)(x^2 -6x + 13). The real root is x = 4; the imaginary roots are 3+2i and 3 - 2i.

Answer 73

But what if an equation is not given. For example, "a cubic function a real zero 4, what is the function"

Answer 74

That question is impossible correct?

Answer 75

Which is what his homework is asking right?

Answer 76

ughhh i give up :(

Answer 77

i understand the first 2 but this last one is confusing

Answer 78

Just given the graph with one x-intercept and no equation, there is no way to write the function, unless you know all the roots, or additional clues to the given polynomial.

Answer 79

ok so i'll just write no solution! :)

Answer 80

That's what I thought. So than if that's what the question asks, just make up 2 other zeros

Answer 81

@pretyroze

What is confusing? I will try to unconfuse you, so that it is crystal clear.

Answer 82

So if you have 4, then (x-4)(x-?)(x-?)

Answer 83

@evan97 Please type your entire question. I dont want to second-guess as to whats bothering you.