Finding Constants A and b in y Abx Using Given Points

When working with exponential functions of the form y Ab^x, where A and b are constants, it is often necessary to determine these constants based on given points through which the graph passes. In this article, we will explore how to find the values of A and b given that the graph passes through the points (2, 2) and (5, 16).

Step-by-Step Solution

Given the points (2, 2) and (5, 16), we can set up the following system of equations:

2 Ab^2 16 Ab^5

By solving these equations, we can find the values of A and b.

Elimination Method

First, we can solve one of the equations for A:

A frac{2}{b^2}

Substituting this expression for A into the second equation:

16 left(frac{2}{b^2}right)b^5 which simplifies to 16 2b^3

Thus, we can solve for b:

b^3 8 b 2

Now that we have the value of b, we can find A by substituting b back into one of the original equations:

2 A(2^2) which simplifies to 2 4A A frac{2}{4} frac{1}{2}

Therefore, the values of A and b are A frac{1}{2} and b 2.

Verification

We can verify that these values satisfy both original equations:

2 left(frac{1}{2}right)(2^2) left(frac{1}{2}right)4 2 16 left(frac{1}{2}right)(2^5) left(frac{1}{2}right)32 16

Visualizing the Graph

To visualize the graph of y frac{1}{2}b^x with b 2, you can use tools like Desmos. By plotting the points and the function, you can see how the graph behaves and confirm the solution using a graphing calculator or software.

Conclusion

In this article, we explored how to find the constants A and b in the exponential function y Ab^x using given points. By solving a system of equations and verifying the solution, we determined that A frac{1}{2} and b 2. This method can be applied to other similar problems involving exponential functions in algebra.