Find a formula for the exponential function passing through the points (-3,5/27) and (1,15).
f(x)=
y = a*b^x
a/b^3 = 5/27
ab = 15
divide, and yo have 1/b^4 = 1/81
b = 3
so, a = 5
y = 5 * 3^x
Find a formula for the exponential function passing through the points (-1,4/5) and (3,500)
Sure, here's a clown equation for you: f(x) = (1/9)^(x+1) + 14
To find the formula for the exponential function passing through the points (-3, 5/27) and (1, 15), we need to use the general exponential function formula:
f(x) = a * b^x
where a represents the initial value of the function and b represents the base of the exponent.
Step 1: Plug in the first point (-3, 5/27) into the formula:
5/27 = a * b^(-3)
Step 2: Plug in the second point (1, 15) into the formula:
15 = a * b^1
Step 3: Rearrange the equations to solve for a and b:
For the first equation, take the reciprocal of both sides to eliminate the negative exponent:
27/5 = b^3
For the second equation, we can rewrite it as:
15/a = b
Step 4: Substitute the value of b from the second equation into the first equation:
27/5 = (15/a)^3
Step 5: Solve for a by simplifying the equation:
27/5 = (15^3/a^3)
27/5 = 3375/a^3
a^3 = 3375 * 5/27
a^3 = 625
a = 5
Step 6: Substitute the value of a into the second equation to solve for b:
15/5 = b
b = 3
Step 7: Write the final formula for the exponential function:
f(x) = 5 * 3^x
To find the formula for the exponential function passing through the points (-3,5/27) and (1,15), we can use the general form of an exponential function: f(x) = a * b^x, where 'a' and 'b' are constants that we need to determine.
Let's start by substituting the coordinates of the first point (-3,5/27) into the equation: 5/27 = a * b^(-3).
Next, substitute the coordinates of the second point (1,15) into the equation: 15 = a * b^1.
We now have a system of two equations:
Equation 1: 5/27 = a * b^(-3)
Equation 2: 15 = a * b
To solve this system, we can isolate either 'a' or 'b' in one equation and substitute it into the other equation.
Let's start by isolating 'a' in Equation 2:
a = 15 / b
Now, substitute this expression for 'a' into Equation 1:
5/27 = (15 / b) * b^(-3)
Next, simplify the equation by multiplying both sides by b^3:
5/27 * b^3 = 15
To solve for 'b', we can multiply both sides by 27 to get rid of the fraction:
b^3 = 15 * 27
Simplify the right side of the equation:
b^3 = 405
To solve for 'b', we can take the cube root of both sides:
b = ∛(405)
Calculating the cube root of 405 gives us:
b ≈ 3.682
Now substitute this value of 'b' back into Equation 2 to solve for 'a':
15 = a * (3.682)
To solve for 'a', divide both sides by 3.682:
a ≈ 15 / 3.682
Calculating this gives us:
a ≈ 4.075
Therefore, the exponential function that passes through the points (-3,5/27) and (1,15) can be written as:
f(x) ≈ 4.075 * (3.682)^x