please help with algebra 2

Question

write an exponential equation given the two points. (1, -4) and (4, -32)

Answer 1

Standard form for an exponential function is f(x) = a·b^x , where a is the y-intercept of the graph (and represents the initial value or amount) and b is the base number, where b > 0, and controls the rate of growth or decay. We have an exponential growth function when b > 1 , and exponential decay when 0 < b < 1.

So what we need is a number for b and a number for a in order to write the equation.

Given two points, we will always solve first for b in the following way:

Using f(4): - 32 = a·b⁴

Using f(1): - 4 = a·b¹

Dividing the top equation by the bottom, left side by left side and right side by right, gives 8 = b³.

So b = 2. Substituting b = 2 into either equation above gives a = -2.

f(x) = - 2·2^x

Btw, this method of solving is analogous to finding geometric means in a geometric sequence:

Find the geometric sequence in which a₁ = -4 and a₄ = - 32: -4 , ____ , ____ , -32 , ...

For a geometric sequence, a_n = a₁·r^n-1 so - 32 = -4·r³ and r = 2.

Answer 2

One way to solve this is to let f(x) = a b^x and then use the two points to find values for a and b

From the second point -32 = a b⁴

from the first point -4 = a b¹

divide the first equation by the second one to get

8 = b³ so b = 2

The second equation then leads to a = -2

Finally f(x) = -2 2^x or f(x) = - 2^1+x