Problem 60 - Algebra 1

Question

Sketch the graph of the exponential equation. $$y=4(1.5)^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(y=4(1.5)^{x}\) starts at the point (0,4) and curves upward as \(x\) increases, representing exponential growth.

Step 1: Identifying the parts of the function

The given function is \(y=4(1.5)^{x}\). Here, the base is 1.5, the coefficient of \(x\) is 1 (since it's not explicitly stated), and no constant is added or subtracted, thereby making 4 the initial value of \(y\). The function, therefore, represents an exponential growth.

Step 2: Setting the x and y-axes

Before plotting the points and sketching the graph, set up an x and y-axes on the graph paper. Indicate the origin, which is the point (0, 0). This becomes the reference point from where the function starts.

Step 3: Plotting the points

Start with x = 0 (for an exponential function, as x approaches -∞, y approaches 0, but never reaches it). So, when \(x=0\), using the function \(y=4(1.5)^{x}\), we get \(y = 4 * 1 = 4\). So, we plot the point (0, 4). Let's choose another point, for example, when \(x = 2\), then \(y = 4(1.5)^2 = 9\). So we plot the point (2, 9).

Step 4: Sketching the graph

Join those points to get a curve and the graph. Extend the line to the x-axis but not crossing it (as the function will never become 0, it will just get closer). Likewise, as x grows positively, y tends to grow toward positive ∞.

Key Concepts

Graphing Exponential Functions

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

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

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.