Final answer:
To find an exponential function passing through (-3, 3/8) and (1, 6), we set up a system of equations using the general form f(x) = ab^x and solve for a and b. By substituting the points into this form, calculating, and manipulating the equations accordingly, we can find the specific exponential function.
Step-by-step explanation:
The student in question is seeking a formula for an exponential function that passes through two specific points: (-3, 3/8) and (1, 6). To find this formula, let's denote the exponential function as f(x) = ab^x, where a is the initial value when x is 0, and b is the base of the exponential function. We are given f(-3) = 3/8 and f(1) = 6, which can be used to create two equations:
- 3/8 = a * b^(-3)
- 6 = a * b^(1)
By solving this system of equations, we find the values of a and b, thus obtaining the formula for the exponential function.
- Assign the given points to our general formula: f(-3) = ab^(-3) = 3/8 and f(1) = ab^(1) = 6.
- With two points, we have two equations. Solve one of these equations for a to find its value in terms of b.
- Substitute the value of a into the other equation to find the value of b.
- Using the value of b, substitute back to find a.
- Write down the exponential equation using the found values of a and b.
To demonstrate the steps involved in the calculation, we can take the logarithm of both sides of one equation to assist in solving for one of the variables. Variables may also be isolated by division or other algebraic methods to solve the system of equations. Remember that working with exponentials and logarithms requires careful manipulation of equations.