Finding a Power Function Through Given Points: A Step-by-Step Guide

When working with mathematical functions, understanding how to find a power function (y ax^b) that passes through specific points can be quite intriguing. In this article, we will explore how to find such a function when it passes through the points (4, 6) and (8, 15). This problem not only involves algebraic manipulation but also demonstrates the powerful use of logarithmic properties.

Step 1: Setting Up the Equations

To find the power function (y ax^b) that passes through the points (4, 6) and (8, 15), we can set up a system of equations using these points. Substituting these points into the equation, we get:

For the point (4, 6): For the point (8, 15):

Expressing the equations, we have:

[[6 a4^b]

[[15 a8^b]

Step 2: Eliminating the Constant 'a'

To eliminate 'a', we divide the second equation by the first equation:

[[frac{15}{6} frac{a8^b}{a4^b}]

Simplifying this expression, we get:

[[frac{15}{6} frac{8^b}{4^b}]

Further simplification yields:

[[frac{5}{2} left(frac{8}{4}right)^b 2^b]

Step 3: Solving for 'b'

Now that we have 'b', we can solve for it by taking the logarithm of both sides:

[[b log_2left(frac{5}{2}right)]

Using a calculator, we find:

[[b approx 1.3219281]

Step 4: Finding 'a'

Substituting 'b' back into the first equation to find 'a', we get:

[[6 a4^{log_2left(frac{5}{2}right)}]

Expressing 4 as (2^2), we get:

[[4^{log_2left(frac{5}{2}right)} (2^2)^{log_2left(frac{5}{2}right)} 2^{2log_2left(frac{5}{2}right)} left(frac{5}{2}right)^2 frac{25}{4}]

Substituting this back, we get:

[[6 aleft(frac{25}{4}right)]

Solving for 'a', we find:

[[a 6 cdot frac{4}{25} frac{24}{25}]

Step 5: Final Equation

Now, substituting 'a' and 'b' back into the power function, we get:

[[y frac{24}{25} x^{log_2left(frac{5}{2}right)}]

This represents the power function that passes through the points (4, 6) and (8, 15).

Alternative Method: Using Logarithmic Properties

Alternatively, we can use the properties of logarithms to find the values of 'a' and 'b'. Taking the natural log of both sides of the power function, we get:

[[ln y ln a b ln x]

Using the two known points (4, 6) and (8, 15), we can set up two linear equations:

[[ln 6 ln a b ln 4]

[[ln 15 ln a b ln 8]

By subtracting the first equation from the second, we find 'b':

[[b frac{ln 15 - ln 6}{ln 8 - ln 4} frac{ln left(frac{15}{6}right)}{ln left(frac{8}{4}right)}]

Calculating the values, we get:

[[b approx 1.3219281]

Substituting 'b' back into the first equation to solve for 'a', we find:

[[ln 6 ln a 1.3219281 ln 4]

Solving for 'a', we get:

[[a e^{ln 6 - 1.3219281 ln 4} approx 0.96]

Thus, the power function is:

[[y 0.96x^{1.3219281}]

Conclusion

In this article, we demonstrated how to find a power function (y ax^b) that passes through specific points. Whether through algebraic methods or the use of logarithmic properties, the process can be both educational and rewarding. Understanding these methods helps in solving various mathematical problems and provides a deeper insight into the nature of power functions and logarithms.