Question

Without graphing, find the domain and range of each absolute-value function.
y = |x - 6| Without graphing, find the domain and range of each absolute-value function.
y = |x - 6| @Mathematics

Answer 1

The domain of y = a|x-h|+k is the set of all real numbers (since you can plug in any real number for x)


The range of y = a|x-h|+k is [k, infinity) when a > 0


Since a = 1 > 0 and k = 0, this means that the range is [0, infinity)

Answer 2

i dont understand

Answer 3

which part doesn't make sense

Answer 4

where did you get a and all that?

Answer 5

y = |x - 6|

Answer 6

the general absolute value equation is


y = a|x-h| + k


In the case of y = |x - 6|, we can rewrite it as y = 1|x - 6|+0


So this shows us that a = 1, h = 6, and k = 0

Answer 7

@fewscrewsmissing 0 is in the range, so it's really [0, infinity)

Answer 8

would that be the case on \[y = |x| - 9\]

Answer 9

domain for y = |x| - 9 is the same (-infinity, infinity)

Answer 10

how?

Answer 11

the range will be different


y = |x| - 9 can be written as y = 1|x-0| + (-9)


So a = 1, h = 0, and k = -9


So the range is [-9, infinity)

Answer 12

The domain is the set of all real numbers since you can plug in any real number for x in y = |x| - 9

Answer 13

so why is -9 the x?

Answer 14

you mean k?

Answer 15

yea,

Answer 16

why is it the first in (-9, infinity) ?

Answer 17

because y = 1|x-0| + (-9) matches up with y = a|x-h|+k

Answer 18

notice the last terms are -9 and k, so k = -9

Answer 19

so k is x?

Answer 20

k is just the vertical shift

Answer 21

confused..

Answer 22

k is just a constant, and it's not a variable like x

Answer 23

y \[y \ge -9\]

Answer 24

right?

Answer 25

yep, that's the range

Answer 26

oh wow, thanks.

Answer 27

hold on

Answer 28

[-9, infinity) is another way of saying \[\Large y \ge -9\]

Answer 29

so if a vertex is (-3,0) the range is \[y \ge0\]

Answer 30

which one are you working on, y = |x| - 9 ???

Answer 31

or a different one

Answer 32

no, this is a different one, i find the vertex, the domain i sall real number and the vertext is -3,0 so is it \[y \ge0\]

Answer 33

ok, if the vertex is (-3, 0), then the range is \[\Large y \ge 0\] so you are correct

**Note: this is assuming that a > 0 **

Answer 34

so the 2nd number in the vertex is always the range?

Answer 35

if the vertex is (h, k) and a > 0, then the range is \[\Large y \ge k\]

Answer 36

so in a way yes, but you have to be careful about the value of 'a'

Answer 37

y = |x - 6| so that range is y\[\ge infinite\]

Answer 38

y = |x - 6| really looks like y = |x - 6| + 0

Answer 39

so k = what?

Answer 40

1

Answer 41

remember the form: y = a|x-h| + k

Answer 42

0

Answer 43

yep k = 0

Answer 44

so that means?

Answer 45

the range is based on k

Answer 46

if a > 0, then the range is \[\Large y \ge k\]

Answer 47

so \[y \ge 0\]

Answer 48

yep

Answer 49

whoa,

Answer 50

lol yep

Answer 51

thanks,.

Answer 52

np

Answer 53

hold on one sec,

Answer 54

sure

Answer 55

why is it when the vertex is -3,0 the range is \[y \ge3\]

Answer 56

the vertex is (h,k)

Answer 57

so k = 0, not 3

Answer 58

or -3

Answer 59

should it be \[y \ge0\]

Answer 60

yes it should be y >= 0

Answer 61

but on the example its

Answer 62

• the vertex is (-2, 0)

Answer 63

• the range is described by y ≥ 0.

Answer 64

range stays the same since k is still 0

Answer 65

shouldnt it just be the 2nd number?

Answer 66

what do you mean

Answer 67

on the vertex its -2,0 so the range is described by y ≥ 0. 0 is the 2nd number

Answer 68

yes the first number of the vertex does not affect the range at all

Answer 69

think of it graphically: the first number just shifts the graph side to side and it does not move it up and down

Answer 70

so if iy was the vertex of (3,0) it would be y\[y \ge0\]

Answer 71

yep, if the vertex was (223, 0), the range would still be the same

Answer 72

now back to this y=|x-6|

Answer 73

how do we find the range?

Answer 74

y=|x-6| is the same as y=|x-6|+0

Answer 75

you said its 0,infinity

Answer 76

yep [0, infinity)

Answer 77

so its \[y \ge infinity\]

Answer 78

which is the same as \[\Large y \ge 0\]

Answer 79

oh!!

Answer 80

wow,

Answer 81

no it's not greater than infinity, that doesn't make sense (and isn't possible)

Answer 82

yea,

Answer 83

the first is in interval notation

Answer 84

the second is a simple inequality

Answer 85

hold on so its y=|x-6| + 0

Answer 86

yep, that's another way of saying y = |x-6|

Answer 87

so where do we get the infinity from? in 0, infinity

Answer 88

is it always gonna be infinity?

Answer 89

when we say \[\Large y \ge 0\], we're saying that y can be any number larger than 0 (or it could equal 0)


[0, infinity) is another way of expressing this interval


Basically [0, infinity) is describing all the values of y by using the endpoints (ie 0 is the smallest y can be and infinity is the largest it can be)

Answer 90

no cause y+ |x-6| would be a=1 h=6 and k=0

Answer 91

y= *********

Answer 92

so wouldnt it be 0,6 then the range is \[y \ge 6\]

Answer 93

no, remember we're using k for the range and not h

Answer 94

i thought k was the first number?

Answer 95

so 6,0 ?

Answer 96

the form is y = a|x-h|+k

y = |x-6| really looks like y = 1|x-6| + 0


So h = 6, k = 0

So the vertex is (6, 0)

and the range is \[\Large y \ge 0\]

Answer 97

oh so in y=|x|-9 it would be y=1|x-h|-9 where a =1 h=0 and k=-9 so it's 0,-9 so tyhe range is \[y \ge-9\]

Answer 98

yep, you nailed it