Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function.\(f(x)=2 e^{0.12 x}\)

Step-by-Step Solution

Answer

The graph of the exponential function \(f(x)=2e^{0.12x}\) starts at the point \((0,2)\) and increases as \(x\) increases. It approaches, but never reaches, the x-axis.

Step 1: Identify the key features of the function

Here, \(a=2\) and \(k=0.12\). So, when \(x=0\), the graph will start from \((0,2)\). The function is an increasing function as \(k > 0\).

Step 2: Identify other key points

To further illustrate the increasing nature of the function, you could choose some points for \(x\) and calculate \(f(x)\). For instance, take \(x=5, 10, 15\), and compute \(f(x)\) at these values. They will be \((5,3.22)\), \((10,4.37)\), \((15,5.93)\), respectively. This will give you some points to plot on the graph.

Step 3: Draw the Graph

Plot these points on a graph and sketch a smooth curve passing through these points. The graph should start from \((0,2)\), and as \(x\) increases, \(f(x)\) should show an increasing trend. Remember, the graph will never touch the x-axis as the exponential function is always above \(0\) for real values of \(x\).

Key Concepts

Increasing Function

An increasing function is one where, as the input value (in this case, \(x\)) increases, the output value \(f(x)\) also increases. For the function \(f(x) = 2e^{0.12x}\), note that 0.12 (the value of \(k\)) is greater than zero.
This positive value of \(k\) indicates that the function will grow as \(x\) increases, thereby making it an increasing function.

• An increasing function means that if you were to draw a line, the line would slope upwards as you move from left to right.
• This growth essentially means for every step you take on the \(x\)-axis, you move higher on the \(y\)-axis.

This is a crucial characteristic of exponential functions that have a positive rate (\(k\) > 0). Such functions are widely used to represent real-life scenarios like population growth or interest calculations where the number continually rises.

Exponential Growth

Exponential growth refers to a situation where the rate of growth is proportional to the current value, resulting in the function growing faster as time passes. For the function \(f(x) = 2e^{0.12x}\), exponential growth is exhibited because the base of the exponent, \(e\), is more than 1 and the exponent \(0.12x\) has a positive coefficient.
This exponential nature is shown through:

• Continuous compounding growth. The function value increases rapidly as \(x\) grows.
• When \(x = 0\), the function starts at 2, but even a slight increase in \(x\) causes \(f(x)\) to rapidly increase.

Real-world examples of exponential growth include investments growing at a fixed interest rate or populations increasing rapidly without limits. Understanding this helps in predicting how functions behave over time.

Graph Sketching

Graph sketching is about visually representing a function on a coordinate system. For \(f(x) = 2e^{0.12x}\), begin by plotting key points. Start with the initial value at \((0,2)\) and then use calculated points like \((5,3.22)\), \((10,4.37)\), and \((15,5.93)\).
These points illustrate the exponential growth starting from the y-intercept and moving sharply upwards:

• Starting from point \((0,2)\) shows the function's initial condition.
• The graph swoops upwards as the \(x\) value increases, a hallmark of exponential complexity.

When sketching, ensure the curve never touches the x-axis because exponential values do not become zero. Smoothly connect the dots, maintaining the upward arc representative of the increasing nature alongside exponential growth.

Key Features Identification

Identifying key features of a function before plotting it simplifies understanding how the function behaves. For \(f(x) = 2e^{0.12x}\), several key features emerge:

• Y-intercept: This is where the graph crosses the y-axis, which is at \((0,2)\).
• Asymptote: The x-axis acts as a horizontal asymptote. The graph never reaches or dips below this line.
• Domain: This function is defined for all real \(x\), meaning there are no restrictions on the x-values.
• Range: Because the function is always positive, the range of \(f(x)\) is \((0, \infty)\), never touching zero.

These features provide all necessary insights needed to sketch and understand the behavior of the function, making graph sketching an easier and more intuitive process.