The given function is:

$f(x) = 2 \cdot 3^x$

You want to change a value in this function such that the graph passes through the point $(3, 27)$.

Step 1: Analyze the current function at $x = 3$

Substitute $x = 3$ into the original function:

$f(3) = 2 \cdot 3^3 = 2 \cdot 27 = 54$

The current value of $f(3)$ is 54, but we want it to be 27.

Step 2: Introduce a constant $a$ to modify the function

Let's modify the function by introducing a constant multiplier $a$ instead of 2. So, the new function becomes:

$f(x) = a \cdot 3^x$

Step 3: Solve for $a$ to make $f(3) = 27$

We know that $f(3) = 27$, so substitute $x = 3$ and solve for $a$:

$a \cdot 3^3 = 27$ $a \cdot 27 = 27$ $a = \frac{27}{27} = 1$

Step 4: Final function

Thus, the function should be:

$f(x) = 3^x$

This new function passes through the point $(3, 27)$.

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Would you like further details on how we changed the function, or have any questions about this process?

Additional Questions:

1. How does the base of the exponential function affect the graph's steepness?
2. What happens if you change the exponent in an exponential function?
3. Can the modified function $f(x) = 3^x$ be written in another form?
4. How would the graph change if we had added a constant term, e.g., $f(x) = 3^x + 5$?
5. What are other ways to modify exponential functions?

Tip:

When working with exponential functions, you can often adjust the base or a constant factor to manipulate how the graph shifts or stretches.