Algebra 1b unit 4 PLEASE HELP

Question

Observe the patterns of the functions b(x) and c(x) after plotting the points onto a graph and connecting the points.

x	-3	-2	-1	0	1	2	3
b(x)	1/64	1/16	1/4	1	4	16	64
c(x)	-13/4	-2	-3/4	1/2	7/4	11/4	17/4

b. Which function could be described as exponential and which as linear? Explain.

c. If the functions continue with the same pattern, will the function values ever be equal? If so, give estimates for the value of x that will make the function values equals. If not, explain why the function values will never be equal.

Answer 1

b(x) is exponential because the y values can be found with this function f(x)=4^x so 4^2=16 and 4^-2=1/16 and 4^3=64 and 4^-1=1/4

c(x) is linear but there doesn't seem to be a constant slope because

if 17/4-11/4=6/4 or y2-y1 and 3-2=1 then the slope is 6/4/1 or just 6/4 because of y/x but if we do 11/4 or x2-x1 -7/4 which is 4/4 which is 1 then do 2-1=1 then due to the slope being y/x we get 1/1 or just 1 showing that the differences between two points are not constant so if anything c(x) is a linear equation but not a constant one because if 5 is added to the numerator then you can see that adding 5 to the numerator of -13/4 will result in -8/4 which is -2 and if you add 5 to -8 in -8/4 you get -3/4 and so on but it seems that 11/4 is off since 7/4 when adding 5 to its numerator becomes 12/4 and when adding 5 to 12 you get 17/4 so there must be an error in the data set. In regards to these two functions never being equal it seems that the rate of increase is too great to warrant any chance of equality because c(x) is increasing by a factor of 5 while b(x) is increasing by a factor of 4^x and it is known that exponential functions have much greater values as the x values increase towards the positive x-axis while linear equations have a steady increase that doesn't grow as much as exponential equations simply because multiplying a number by itself increasing it faster than simply adding a number by itself so 4 times 4 will be 16 while 4+4=8 and 9 times 9 is 81 while 9+9=18 so of the functions continue with the same pattern of increase the values of the outputs will never be equal because exponential functions will always increase in value in a short time while linear equations will increase in value in a long time.

Answer 2

well notice, that b(x) QUADRUPLES when x increases by 1,

which suggests EXPONENTIAL

That would suggest B(x) = 4^x....

in fact, 4^(-3) = 1/64, 4^(-2) = 1/6, 4^(-1) = 1/4, 4^0 = 1, 4^1=4 4^2=16, and 4^3 = 64

With the exception of x=2,3, B(x) increases by 5/4 when x increases by 1..

(are you sure that is not a typo????)

that would suggest LINEAR with a slope if 5/4...

since B(0) = 1/2 the y-intercept is b=(0,1/2), so

the linear function is y = (5/4)x + 1/2

SO will they meet? If so, then there are two adjacent,consecutive

data points where one function is greater than the other (HIGHER ON THE Y-AXIS)

at one point and then at the next they switch...

unfortunately, based on the data, it does NOT look like c(x) will ever

catch up to b(x); the exponential function will ALWAYS dominate the linear

more precisely, we are interested in solving:

b(x) = c(x)

4^(x) = (5/4)x + 1/2

which can only be solved numerically...

But if b(x)=c(x), then F(x) = b(x)-c(x) = 0

So examining the graph of F(x) = b(x)-c(x)

= 4^(x) - (5/4)x - 1/2

NOWHERE does F(x) cross the x-axis...

so no, they will not meet