Problem 37 - Algebra and Trigonometry

Question

(a) Sketch the graphs of \(f(x)=2^{x}\) and \(g(x)=3\left(2^{x}\right)\) (b) How are the graphs related?

Step-by-Step Solution

Answer

The graph of \(g(x)\) is a vertical stretch of the graph of \(f(x)\) by a factor of 3.

Step 1: Understand the Basic Graphs

First, understand the shape and basic characteristics of the function \( f(x) = 2^x \). This is an exponential function which passes through the point (0,1), increases rapidly, and never touches the x-axis (asymptotic to the negative side of the x-axis).

Step 2: Plot the Graph of \(f(x)=2^x\)

Draw a graph of \( f(x) = 2^x \) starting from a few key points: \( (0, 1)\), \( (1, 2)\), \( (2, 4)\), etc. Note the smooth exponential increase, ensuring the curve approaches the x-axis without touching it as it moves left towards negative infinity.

Step 3: Understanding the Effect of Scaling on \(f(x)\)

The function \( g(x) = 3(2^x) \) is derived from \( f(x) = 2^x \) by multiplying by 3. This means at each point \( x \), the value of \( f(x) \) is scaled (or stretched vertically) by a factor of 3.

Step 4: Plot the Graph of \(g(x)=3(2^x)\)

Based on the transformation understanding, draw the graph of \( g(x) = 3(2^x) \) by multiplying the f(x) values you plotted earlier by 3. Key points include \( (0, 3)\), \( (1, 6)\), \( (2, 12)\), etc.

Step 5: Compare and Analyze the Graphs

Compare the graphs of \( f(x) \) and \( g(x) \). Notice that g(x) is a vertical stretch of f(x) by a factor of 3. Both graphs have the same shape and horizontal asymptote; however, g(x) reaches higher y-values at each x.

Key Concepts

Exponential Growth

Exponential growth is a fascinating mathematical concept that describes how a quantity increases at a consistent multiplicative rate over time. Imagine you are looking at the graph of the basic function \( f(x) = 2^x \). This function represents exponential growth, showcasing how, starting from an initial value, the function

• Doubles each time \( x \) increments by 1.
• Has a J-shaped curve that grows increasingly steeper as \( x \) increases.
• Never crosses the x-axis, embracing this line asymptotically towards the left.

Exponential growth is evident in everything from population dynamics to compound interest. The pattern is that each step sees growth by a consistent multiplication factor.

Vertical Stretching

Vertical stretching is a type of transformation we apply to functions that affects their height, rather than their breadth. Let's take a closer look at the function \( g(x) = 3(2^x) \). This is a vertically stretched version of \( f(x) = 2^x \).

Here's how vertical stretching works:

• We multiply the original function's value, in this case \( f(x) = 2^x \), by a constant.
• That constant is 3 for the function \( g(x) \), indicating a threefold increase or stretch in its height.
• The shape of the graph remains the same, but every point is now thrice as high as in \( f(x) \).

This transformation doesn't affect the horizontal features like the asymptotic behavior, only magnifying the y-values of the graph.

Function Transformation

Function transformation is a powerful tool in mathematics that allows us to modify graphs to achieve specific effects. Our graphs of \( f(x) = 2^x \) and \( g(x) = 3(2^x) \) showcase the idea of transformation through vertical stretching.

Here's what transformations can involve:

• Changing their position along the x- or y-axis (translation).
• Altering their steepness or orientation (scaling like vertical stretching or flipping).
• Modifying the graph's fitting with real-world phenomena through changes in growth rate or decay.

In our example, the transformation from \( f(x) \) to \( g(x) \) doesn't move the graph but stretches it vertically, making transformations an essential part of graphing and analyzing functions. By understanding these effects, one can interpret, predict, and visualize how different functions behave under various conditions.