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Chapter 4: Problem 37

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

Short Answer

Expert verified
The graphs of \(f(x)\) and \(g(x)\) are vertically scaled versions of each other, with \(g(x)\) stretched by a factor of 3.

Step by step solution

01

Understanding the Graphs

Before sketching the graphs, we need to understand the functions. The function \(f(x) = 2^x\) represents an exponential growth where the base is 2. The function \(g(x) = 3(2^x)\) is similar to \(f(x)\), but it is multiplied by 3, which affects the vertical stretching.
02

Sketching Graph 1: \(f(x) = 2^x\)

Plot the function \(f(x) = 2^x\). This graph is a typical exponential curve that starts below the x-axis for negative x-values, passes through the point (0, 1), and rises rapidly for positive x-values. It is always increasing and never touches the x-axis, showing exponential growth.
03

Sketching Graph 2: \(g(x) = 3(2^x)\)

Now plot the function \(g(x) = 3(2^x)\). This graph has the same shape as \(f(x) = 2^x\) because it is also an exponential curve. However, it passes through the point (0, 3) because every y-value of \(f(x)\) is multiplied by 3, indicating that \(g(x)\) is a vertically stretched version of \(f(x)\) by a factor of 3.
04

Comparing the Graphs

Notice that both graphs have the same base, so they have the same horizontal growth pattern. The graph of \(g(x) = 3(2^x)\) is a vertical stretch of the graph of \(f(x) = 2^x\) by a factor of 3. This means every point on \(f(x)\) is three times higher on \(g(x)\). The two graphs are similar in shape and orientation, but not in size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Transformations
When we discuss graph transformations, we are focusing on how a graph's appearance can change by altering, shifting, or stretching it. For exponential functions like \(f(x) = 2^x\) and \(g(x) = 3(2^x)\), transformations are often visible in the vertical direction. A key transformation at play here is vertical stretching, but it's just one part of graph transformations. Here are a few types of transformations to be aware of:
  • Translation: Moving the graph up, down, left, or right.

  • Reflection: Flipping the graph over an axis.

  • Vertical Stretch/Compression: Changing the height of the graph without altering the basic shape.
In the case of our functions, \(f(x)\) and \(g(x)\), it is mainly vertical stretching that we observe, as detailed in the vertical stretching section.
Exponential Growth
Exponential growth describes a situation where a quantity increases by a consistent percentage over equal intervals. For a function like \(f(x) = 2^x\), this is visually manifested as a curve that gets steeper as \(x\) gets larger. The base of the exponent, 2, means each step increases by a factor of 2. Here's why it's crucial:
  • Dynamic Increase: There's no cap; values keep doubling, rising sharply on the graph.

  • Real-World Examples: Population growth and compound interest are often modeled as exponential functions.
Reading the graph of \(f(x) = 2^x\), observe how it passes through the point (0,1) and rapidly increases for positive \(x\) values. That's the hallmark of exponential growth. It never touches the x-axis and continues to rise as \(x\) grows.
Vertical Stretching
Vertical stretching is a type of graph transformation that enlarges a graph in the vertical direction by multiplying all y-values by a constant factor. For instance, given \(f(x) = 2^x\) and its stretched form \(g(x) = 3(2^x)\), the number 3 causes each point on \(f(x)\) to be scaled up by three times its original height. Here's how to identify and analyze vertical stretching:
  • Effect on Graph: While the shape and direction of the graph remain the same, the function reaches greater heights.

  • Specific Changes: If \(f(0) = 1\) for \(f(x) = 2^x\), then \(g(0) = 3 \, \times \, 1 = 3\) for \(g(x) = 3(2^x)\).
This means that every point on \(g(x)\) is simply three times higher than \(f(x)\). Despite this vertical shift, the basic properties of the graph, such as its increasing nature and asymptotic behavior towards the x-axis, stay unchanged.

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