Question

HELPPP

How can you solve the equation 4^-x = 2^x + 3 graphically?

Answer 1

Draw the graphs of y = 4 - x and y = 2x +3 on the same set of axes and see where they intersect. @maria_brianna

Answer 2

Do you need to draw the graphs or is there more to this problem?

Answer 3

yes it asks to make tables to find the solution to 4^-x =2^x+3 and to take the integer values of x between -3 and 3 @Directrix

Answer 4

is it
\[ 4^{-x} = 2^{x+3} \]??

Answer 5

yess

Answer 6

@phi

Answer 7

they want you to find the "y" values for the equation
\[ y = 4^{-x} \]
and
the equation
\[y= 2^{x+3} \]

Answer 8

for x= -3, -2, -1, 0 , 1 , 3

I got that from
***make tables to find the solution to 4^-x =2^x+3 and to take the integer values of x between -3 and 3***

Answer 9

for the first equation
\[ y = 4^{-x}\]
do you know how to figure out y when x is -3 ?
you replace the x with -3:
\[ y= 4^{-(-3)} \]
minus a minus is a plus, so that simplifies to
\[ y= 4^3\]
can you figure out 4^3 (= 4*4*4, or use ^ on your calculator)
what do you get ?

Answer 10

i got 64

Answer 11

now make a table with two columns. the left column is labeled x, the right col is y
in the x column, put in -3, and next to it in the y column put 64

Answer 12

now try to figure out y when x is -2
can you do that ?

Answer 13

i got 0.0625

Answer 14

can you show the steps?

Answer 15

i just typed 4^-2 in my calculater

Answer 16

oh. that is close. but the equation is 4^(-x)

Answer 17

oh my bad its 16

Answer 18

ok. put -2, 16 in your table
now do x=-1
which means do 4^(-(-1)) which is 4^1 (which you can do in your head, right ?)

Answer 19

its 4

Answer 20

now x=0
you can do this in your head, because anything to the zero power is ??

Answer 21

1

Answer 22

now do x=1, 2, 3
we will now start getting fractions (or decimals)

Answer 23

i got the same numbers tho

Answer 24

are you forgetting the equation uses -x (not just x)?

Answer 25

oh lol yea

Answer 26

for 1 got 0.25, for 2 i got 0.0625, for 3 i got 0.015625

Answer 27

yes, or 1/4, 1/16 , 1/64

now you should have the table they want for the first equation.

now we do the same thing for the other equation
\[ y = 2^{x+3} \]

Answer 28

what y do you get for x= -3
?

Answer 29

i got 64

Answer 30

start by replace x with -3 in the equation

Answer 31

so you get
\[ y= 2^{-3+3} \]
where I live, -3+3 is 0

Answer 32

and 2^0 is ??

Answer 33

1

Answer 34

so in this new table (label it the y= 2^(x+3) table)
you get x=-3, y =0

ok ?

Answer 35

now find y when x= -2

Answer 36

so theres gonna be 2 tables?

Answer 37

yes

Answer 38

so for starters of the table it would be the same numbers
?? -3 between 3?

Answer 39

the same x numbers (the left column)
the right column will have different y numbers (because it's a different equation)

the idea is that somewhere (at some x) the y values are the same for both equations at that x. that is the spot we are trying to find.

Answer 40

if the curves never cross we won't find a solution.
but they are being nice... they are more or less telling us to expect the curves to cross somewheres between x=-3 and x=3

Answer 41

oh okk

Answer 42

now find y when x= -2

Answer 43

i got 3.25

Answer 44

can you show your steps?

Answer 45

i plugged 2^-2+3 in the calculator

Answer 46

order of operations means the calculator did exponents first
it did
\[ 2^{-2} \]
and then added 3
but you want to do
\[ 2^{-2+3} \]
which is a different thing. first do the -2+3
you can put parens in for the calculator. or first figure out -2+3, and use that number
for the exponent.

Answer 47

so its just 1?

Answer 48

for the exponent
y= 2^(-2+3) = 2^1 = ??

Answer 49

oh 2

Answer 50

so add -2, 2 to your table


now do x= -1

Answer 51

wait what was the one for x collom -3 ?

Answer 52

y= 2^(-3+3) = 2^(?)

Answer 53

1

Answer 54

for x=-3 you get 2^0 = 1

Answer 55

and -1=4

Answer 56

I would say -1, 4 (the numbers are not equal, but -1 and 4 "go together" )

Answer 57

ok

Answer 58

and for x=-2 you found y= 2
-2, 2

now do x= 0 (getting back on track, I hope)

Answer 59

its 8?

Answer 60

yes
2^(0+3) = 2^3 = 2*2*2 = 8
so 0, 8 are the entries in your table for x, y

now x=1

Answer 61

x=1 is 16 x=2 is 32 x=3 is 64

Answer 62

yes.

now we look at the two tables.
at x=-3 the first equation's y is "above" (bigger) than the y value of the 2nd equation.

Answer 63

at x=3 the first equation's y is "below" (less than) the y value of the 2nd equation.

this means the equations "crossed" somewhere. any idea what x it was?

Answer 64

no

Answer 65

look at your two tables.

do you see the y's getting smaller in the first table?
do you see the y's getting bigger in the 2nd table (as we move from x=-3 to x=+3)?
where are the y's the same?

Answer 66

ohhhhhh theyre the same at -1

Answer 67

which means
\[ 4^{-x} = 2^{x+3} \]
is true at x= -1
we solved the problem!

Answer 68

yaayy thank you so much!!

Answer 69

most times, we will find the point where the first equation's y is bigger than the 2nd's and at the next x , the y value is smaller. that means the "answer" is an x in between the two we used. We would have to work harder, looking for the answer in between those two x's
but here, it was easy (thank goodness)

I hope some of this was helpful.
Good luck!

Answer 70

it was thanks!!