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Graphing exponential functions is a task that involves showcasing the nature of rapid change. An exponential function can be expressed in the form of \(y = a(b)^x\), where \(a\) is the initial value, and \(b\) is the base.

In the equation \(y = 4(1.5)^x\), the graph will depict how the value of \(y\) changes exponentially as \(x\) increases. To graph such functions, start by identifying key characteristics:

• **Base**: Indicates whether the function represents growth or decay. In \(y = 4(1.5)^x\), the base 1.5 is greater than one, signifying **exponential growth**.
• **Initial value**: The coefficient \(a\), which is 4, determines the starting value of the graph when \(x=0\).
• **Asymptote**: Most exponential graphs approach a horizontal asymptote. For positive bases greater than one, this asymptote is usually the \(x\)-axis which the graph nears but never touches.

To sketch the curve, first, set up your axes and label them. Then, mark the points by solving the function for several \(x\) values. Finally, draw a smooth curve that passes through these points, reflecting the rapid increase of the function for positive \(x\) values.

Exponential growth refers to an increase that becomes more rapid in proportion to the growing total number or size. It's a key feature of exponential functions with a base greater than one.

In our function \(y = 4(1.5)^x\), exponential growth is seen as the output value (\(y\)) increases at a fast rate for every incremental increase in \(x\). This is because:

• Every increase in \(x\) multiplies the previous \(y\) value by the base, here 1.5.
• As a result, small initial increases can lead to much larger values later on.

This behavior is crucial in areas like finance, where compound interest uses exponential growth, or populations studies, where organism numbers may increase exponentially under ideal conditions.

Plotting points on a graph is a fundamental step in visualizing mathematical functions, such as exponential equations. You will usually start at \(x = 0\) and then choose subsequent values of \(x\) to see how \(y\) changes.

For \(y = 4(1.5)^x\), when \(x = 0\), the equation simplifies to \(4 \times 1 = 4\). This point is (0, 4). Choose a few more \(x\) values to find more points; for example, \(x = 2\) gives \(y = 9\), yielding the point (2, 9).

By connecting these points smoothly, we portray the function's curve, which shows how the graph ascends more steeply as \(x\) increases. When plotting:

• Ensure axes are well-labeled for clarity.
• Pick a range of \(x\) values, both positive and possible negative for a comprehensive view.
• Keep the curve smooth and continuous, as exponential functions represent continuous growth.

This step is crucial for understanding the behavior of the function and its real-world implications.