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The given problem involves an exponential function of the form:

\[
y = a \times b^x
\]

From the graph, we have two points: \((2, 27)\) and \((3, 54)\).

We can substitute these points into the exponential equation to form two equations:

1. \( 27 = a \times b^2 \)
2. \( 54 = a \times b^3 \)

### Step 1: Divide the equations to eliminate \(a\).

Dividing equation (2) by equation (1):

\[
\frac{54}{27} = \frac{a \times b^3}{a \times b^2}
\]

This simplifies to:

\[
2 = b
\]

### Step 2: Solve for \(a\).

Now substitute \(b = 2\) into one of the original equations. Let's use equation (1):

\[
27 = a \times 2^2
\]

This simplifies to:

\[
27 = a \times 4
\]

Solving for \(a\):

\[
a = \frac{27}{4} = 6.75
\]

### Final Answer:

\[
a = 6.75, \quad b = 2
\]

Would you like a further explanation or have any questions?

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Here are 5 related questions to extend your learning:

1. What is the significance of the base \(b\) in an exponential function?
2. How would the graph change if the base \(b\) was less than 1?
3. Can an exponential function have a negative base?
4. What is the effect of changing the value of \(a\) in an exponential equation?
5. How can you use logarithms to solve exponential equations more generally?

**Tip:** For exponential functions, the value of \(b\) determines the rate of growth or decay. If \(b > 1\), the function grows; if \(0 < b < 1\), the function decays.

The graph below is exponential, and its equation is in the form y = a * b^x. Given two points (2, 27) and (3, 54), find the values of a and b.

Solve the exponential equation y = a * b^x using two points (2, 27) and (3, 54). Learn step-by-step how to find the values of a and b, which define the growth of the function. Suitable for students in Grades 9-12.

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