Vertical Stretch Factor

Definition

The vertical stretch factor, also known as the amplitude, is a parameter that determines the vertical scaling of the graph of an exponential function. It affects the steepness and the range of the function's values, influencing the overall shape and appearance of the graph.

5 Must Know Facts For Your Next Test

1. The vertical stretch factor, $a$, in the exponential function $f(x) = a ullet b^x$ determines the vertical scaling of the graph.
2. A vertical stretch factor greater than 1 ($a > 1$) results in a steeper, narrower graph, while a vertical stretch factor less than 1 ($0 < a < 1$) results in a flatter, wider graph.
3. The vertical stretch factor affects the range of the function's values, with a larger $a$ leading to a greater vertical range and a smaller $a$ leading to a smaller vertical range.
4. The vertical stretch factor can be used to model real-world phenomena, such as population growth, radioactive decay, and compound interest, where the rate of change is proportional to the current value.
5. Understanding the impact of the vertical stretch factor is crucial for accurately interpreting and analyzing the behavior of exponential functions in various applications.

Review Questions

• Explain how the vertical stretch factor affects the graph of an exponential function.
  • The vertical stretch factor, $a$, in the exponential function $f(x) = a ullet b^x$ determines the steepness and range of the graph. A vertical stretch factor greater than 1 ($a > 1$) results in a steeper, narrower graph, while a vertical stretch factor less than 1 ($0 < a < 1$) results in a flatter, wider graph. The vertical stretch factor directly impacts the range of the function's values, with a larger $a$ leading to a greater vertical range and a smaller $a$ leading to a smaller vertical range. Understanding the effect of the vertical stretch factor is crucial for accurately interpreting and analyzing the behavior of exponential functions in various applications.
• Describe how the vertical stretch factor can be used to model real-world phenomena involving exponential growth or decay.
  • The vertical stretch factor, $a$, in the exponential function $f(x) = a ullet b^x$ can be used to model various real-world phenomena where the rate of change is proportional to the current value. For example, in population growth models, the vertical stretch factor represents the initial population size, and the base $b$ represents the growth rate. In radioactive decay models, the vertical stretch factor represents the initial amount of radioactive material, and the base $b$ represents the decay rate. Similarly, in compound interest calculations, the vertical stretch factor represents the initial investment, and the base $b$ represents the interest rate. By adjusting the vertical stretch factor, these models can be tailored to accurately represent the specific characteristics of the real-world scenario being studied.
• Analyze the relationship between the vertical stretch factor and the asymptotic behavior of an exponential function's graph.
  • The vertical stretch factor, $a$, in the exponential function $f(x) = a ullet b^x$ is directly related to the asymptotic behavior of the graph. Asymptotes are straight lines that the graph of an exponential function approaches but never touches, representing the upper or lower limit of the function's values. When the vertical stretch factor $a$ is greater than 0, the graph of the exponential function will have a horizontal asymptote at the $y$-axis. The value of the vertical stretch factor $a$ determines the $y$-coordinate of this asymptote, with a larger $a$ resulting in a higher asymptote and a smaller $a$ resulting in a lower asymptote. Understanding the relationship between the vertical stretch factor and the asymptotic behavior is crucial for accurately interpreting and analyzing the properties of exponential functions in various applications.

Related terms

Exponential Function: A function of the form $f(x) = a ullet b^x$, where $a$ is the vertical stretch factor and $b$ is the base of the exponential.

Horizontal Shift: The horizontal translation of the graph of an exponential function, determined by the $x$-intercept or the parameter $h$ in the function $f(x) = a ullet b^{(x-h)}$.

Asymptote: A straight line that a graph of an exponential function approaches but never touches, representing the upper or lower limit of the function's values.