find the exponential function of the form y=ab^x whose graph passes through the points (4,6) and (7,10)
i got a=3.04 and b= 1.185 so my equation is y=3.04(1.185)^4
is this correct?
one more problem. find a power function of the form y=ax^b whose graph passes through the points (2,3) and (10,21)?
this one im quite confused on how to do, can someone show the steps for it and lead me through until the answer? thanks!
I agree with your a and b values, but your equation should have an exponent of x , not 4
(probably just a typo)
2nd:
for (2,3) --> 3 = a(2^b)
for (10,21) ---> 21 = a(10^b)
divide them
21/3 = 10^b/2^b
7 = (10/2)^b
7 = 5^b
take log of both sides
log 7 = log (5^b)
log 7 = b log 5
b = log 7/log 5 = appr. 1.209
sub back into 1st equation
a(2^1.206) = 3
a(2.3119) = 3
a = 1.298
so y = 1.298(x^1.209)
check my arithmetic on the last one
To check if your first answer is correct, substitute the points (4,6) and (7,10) into your equation and see if they satisfy the equation:
For the point (4,6):
6 = 3.04(1.185)^4
For the point (7,10):
10 = 3.04(1.185)^7
Solving these equations will confirm if your values for a and b are correct.
Now, let's move on to the second problem of finding a power function.
We are looking for a function of the form y = ax^b that passes through the points (2,3) and (10,21).
To find the values of a and b, we can use the following steps:
Step 1: Plug in the coordinates of the first point (2,3) into the equation y = ax^b:
3 = a(2)^b
Step 2: Plug in the coordinates of the second point (10,21) into the equation:
21 = a(10)^b
Now, we have two equations with two variables (a and b):
3 = a(2)^b
21 = a(10)^b
To solve this system of equations, divide the second equation by the first equation:
21 / 3 = (a(10)^b) / (a(2)^b)
7 = (10/2)^b
7 = 5^b
To solve for b, take the logarithm of both sides:
log(7) = log(5^b)
log(7) = b * log(5)
Solve for b:
b = log(7) / log(5)
Now that we have the value of b, we can substitute it back into either of the original equations to solve for a. Let's use the first equation:
3 = a(2)^b
Plug in the value of b, that we just found, into the equation and solve for a:
3 = a(2)^(log(7) / log(5))
Simplify the equation and solve for a:
3 = a(2^(log(7)) / 2^(log(5)))
a = 3 * (2^(log(5)) / 2^(log(7)))
Evaluate these logarithmic expressions using a calculator to get the value of a.
Once you have the values of a and b, substitute them back into the original equation y = ax^b to get your final power function.