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Magazine Chapter 4: Problem 102
Describe the differences in the graphs of \(f(x)=3^{x}\) and \(g(x)=x^{3}\)
Short Answer
Expert verified
The graph of \(f(x) = 3^x\) grows exponentially with an asymptote at y = 0, while \(g(x) = x^3\) grows polynomially and symmetrically around the origin.
Step by step solution
01
Identify and understand the functions
The first function is an exponential function: \(f(x) = 3^x\). The second function is a cubic polynomial function: \(g(x) = x^3\).
02
Identify the key features of the exponential function
For \(f(x) = 3^x\), key features include: 1. The graph passes through the point (0, 1) because \(3^0 = 1\). 2. The graph increases rapidly as x increases. 3. The graph approaches the x-axis but never touches it as x decreases (asymptote at y = 0).
03
Identify the key features of the cubic function
For \(g(x) = x^3\), key features include: 1. The graph passes through the origin (0, 0) because \(0^3 = 0\). 2. The graph increases and decreases symmetrically around the origin. 3. As x approaches positive or negative infinity, \(g(x)\) increases or decreases without bound.
04
Compare the behavior of the functions at specific points
Plugging in values: - For \(x = 2\), \(f(2) = 3^2 = 9\), while \(g(2) = 2^3 = 8\). - For \(x = -2\), \(f(-2) = 3^{-2} = \frac{1}{9}\), while \(g(-2) = (-2)^3 = -8\).
05
Analyze the end behavior
As \(x\) approaches infinity: - \(f(x)\) grows exponentially and positive without bound. - \(g(x)\) grows polynomially but also positive without bound. As \(x\) approaches negative infinity: - \(f(x)\) approaches zero but remains positive.- \(g(x)\) decreases without bound (becomes more negative).
06
Describe the differences in the graph shapes
The graph of \(f(x) = 3^x\) is an increasing curve that approaches an asymptote at y = 0 and passes through (0,1). The graph of \(g(x) = x^3\) is a symmetric curve passing through the origin, increasing to the right and decreasing to the left.
07
Summarize the differences
In summary, \(f(x) = 3^x\) grows exponentially, has an asymptote at y = 0, and only positive outputs for all \(x\). \(g(x) = x^3\) grows polynomially without bounds, has both positive and negative outputs, and is symmetric around the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Function
An exponential function is a mathematical function of the form \(f(x) = a^x\), where \(a\) is a constant and \(a > 0\).
One key feature of exponential functions is their rapid growth or decay. Whether they grow or decay depends on whether \(a > 1\) or \(0 < a < 1\).
For the specific function \(f(x) = 3^x\), the graph grows rapidly as \(x\) increases because \(3 > 1\).
One key feature of exponential functions is their rapid growth or decay. Whether they grow or decay depends on whether \(a > 1\) or \(0 < a < 1\).
For the specific function \(f(x) = 3^x\), the graph grows rapidly as \(x\) increases because \(3 > 1\).
- The curve passes through the point (0, 1) since \(3^0 = 1\).
- It increases quicker than polynomial functions as \(x\) becomes larger.
- As \(x\) decreases, \(3^x\) gets closer to 0 but never touches the x-axis. This behavior is known as approaching an asymptote at y = 0.
Cubic Polynomial
A cubic polynomial is a mathematical function of the form \(g(x) = ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are constants. The function \(g(x) = x^3\) is a simple cubic polynomial where \(a = 1\) and \(b = c = d = 0\).
Here are some key features:
Here are some key features:
- The graph passes through the origin (0, 0) since \(0^3 = 0\).
- It is symmetric with respect to the origin, meaning for every point (x, y), there is a corresponding point (-x, -y).
- As \(x\) becomes very large or very small, the value of \(x^3\) becomes very large or very negative, respectively. This is known as the end behavior of the function.
- The graph increases without bound as \(x\) moves to positive infinity and decreases without bound as \(x\) moves to negative infinity.
Asymptote
An asymptote is a line that a graph approaches but never touches or crosses. It represents the behavior of a function as the input values get very large or very small.
- For the exponential function \(f(x) = 3^x\), as \(x\) approaches negative infinity, \(f(x)\) gets closer and closer to the x-axis (y = 0) but never actually reaches it. This line y = 0 is the asymptote of the graph.
- In other contexts, asymptotes can be vertical, horizontal, or even slant. They provide vital information about the behavior of functions, especially as inputs grow large or small.
- Unlike exponential functions, the cubic polynomial \(g(x) = x^3\) does not have an asymptote. Its graph continues to increase or decrease without bound as \(x\) approaches infinity or negative infinity.
