Question

Write an exponential function \(y=a b^x\) whose graph passes through the given points. \((3,1),(5,4)\)

Short answer

The exponential function that passes through the points \((3,1)\) and \((5,4)\) is \(y=(1/8) \cdot 2^x\).

Step-by-step solution

Substitute the given points into the equation

Given the points \((3,1)\) and \((5,4)\), they can be written in the form \((x_1, y_1)\) and \((x_2, y_2)\) respectively. These points are substituted into the exponential function \(y= a \cdot b^x\) to form two equations \[1=a \cdot b^3\] and \[4=a \cdot b^5\].

Divide the two equations to solve for b

By dividing the second equation by the first, the solution for \(b\) can be obtained \[b= ((4/a)/(1/a))^\frac{1}{2}= 2 \] since \(a\)'s cancel out and because the exponent results from the property of exponents, \(b^{m/n}= (b^m)^{1/n}\).

Substitute b into any of the equations to solve for a

Substituting the value of \(b\) into the first equation gives \[1=a \cdot 2^3\], which simplifies to \(a = 1/8\).

Key concepts

Substituting Points

To start solving a problem involving an exponential function like \(y = a \cdot b^x\), you first need to use the points provided by the exercise. These points are placeholders for \(x\) and \(y\) in your function. By substituting these values into the equation, you form new equations that will help you solve for unknown variables.

In our example, we have the points \((3, 1)\) and \((5, 4)\). These substitute into the function to form the equations \(1 = a \cdot b^3\) and \(4 = a \cdot b^5\).

The process of substituting these points sets the stage for solving the variables—essentially, giving us a system of equations we can work with. This is an essential first step in tackling the most challenging parts of exponential functions.

Solving for Variables

Once you have equations from substituting points, the next step is to isolate and solve for the unknown variables in the function, typically \(a\) and \(b\). One effective strategy is to employ division between the equations if they share a common variable.

For our equations \(1 = a \cdot b^3\) and \(4 = a \cdot b^5\), dividing the second equation by the first allows one to eliminate \(a\), leading to \(b = \left(\frac{4/a}{1/a}\right)^{1/2} = 2\).

This division works because the principles of algebra and exponents—solving for one variable in terms of another—are utilized to simplify the expressions, making the path clearer toward finding the values of all unknowns. Solving exponential functions often requires these logical steps.

Properties of Exponents

Understanding the properties of exponents is crucial when working with exponential functions. These properties simplify complex expressions and help us reach solutions more effectively.

When we divide the equation \(4 = a \cdot b^5\) by \(1 = a \cdot b^3\), we explore a key property that allows us to manage the exponents: \(b^{m-n} = b^m / b^n\). In this case, \(b^5 / b^3 = b^2\).

Using these properties can drastically simplify our calculations. For instance, knowing that \((b^m)^{1/n} = b^{m/n}\) is helpful when taking roots or powers. It ensures that we can handle the often repeated and large numbers that we face in these equations effortlessly. Recognizing and applying these rules is fundamental in solving exponential functions correctly and efficiently.