Question

Write an exponential equation in the form y=ab^x whose graph passes through points (4,8) and. (6,32)

Answer 1

Do you have the time to do it analytically?

Form of exponential equation you desire: y = a * b^(x),

where "a" and "b" are unknown values which we will calculate from a system of two equation based on your Cartesian (x, y) coordinate points.

System of equations:

Equation 1: 8 = a * b^(4) ... from the point: (4, 8)

Equation 2: 32 = a * b^(6) ... from the point: (6, 32)

# I arbitrarily labeled the equations, so we can know what's what. I would recommend canceling out variable "a: by dividing the equations like so ... Equation 2 ÷ Equation 1 = _____???

# If you do it the other way, you will have to resolve a negative exponent on the "b" variable.

32 / 8 = [a * b^(6)] / [a * b^(4)]

# Now, a's obviously cancel by division:

4 = b^(6 - 4) ... # simplifying exponents:

4 = b^(2)

# Calculate the square root, to cancel the squaring:

±√4 = √[b^(2)]

# We will just take the positive solution for "b". That will be good enough.

2 = b

# Now calculate variable "a" from either of the equations:

8 = a * b^(4) ... knowing the value of "b":

8 = a * (2)^(4)

8 = a * 16

# Now if you just divide both sides of the equation by 16, you get the reduced value of "a".

1 / 2 = a ... or 0.5 = a

Solution set: a = 0.5 ; b = 2

Exponential curve which fits the points: y = 0.5 * 2^(x)