Given the table, write the equation of the exponential function in y = a(b)^x form.

Understand the Problem

The question is asking to analyze the given table of values to derive the equation of an exponential function in the form y = a(b)^x. This involves determining the parameters a and b based on the values provided in the table.

Answer

The equation of the exponential function is $ y = 2(4)^x $.

Steps to Solve

Identify Values from the Table

Extract the pairs of $(x, y)$ from the table:

For $x = 1$, $y = 8$
For $x = 2$, $y = 32$
For $x = 3$, $y = 128$
For $x = 4$, $y = 512$
For $x = 5$, $y = 2048$

Find the Ratio of Successive $y$ Values

Calculate the ratio $\frac{y_{n}}{y_{n-1}}$ for $x = 2$ to $5$:

For $x = 2$: $\frac{32}{8} = 4$
For $x = 3$: $\frac{128}{32} = 4$
For $x = 4$: $\frac{512}{128} = 4$
For $x = 5$: $\frac{2048}{512} = 4$

This shows that the common ratio $b = 4$.

Determine the Value of $a$

Using the equation $y = a(b)^x$, substitute for $(x, y) = (1, 8)$:

[ 8 = a(4)^1 ]

Solving for $a$:

[ 8 = 4a \quad \Rightarrow \quad a = \frac{8}{4} = 2 ]

Write the Exponential Equation

Now that we have both $a$ and $b$, we can put them into the exponential equation.

The equation is:

[ y = 2(4)^x ]

The equation of the exponential function is ( y = 2(4)^x ).

More Information

This exponential function describes a growth pattern where $y$ quadruples for every unit increase in $x$. The base $b = 4$ indicates rapid growth.

Tips

Forgetting the base: It's common to miscalculate or overlook the base $b$ when determining the exponential function.
Incorrectly solving for $a$: Ensure to use the correct corresponding $y$ value for the chosen $x$.

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