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Magazine Chapter 27: Problem 10
Sketch the graphs of the given functions. Check each by displaying the graph on a calculator. $$y=4 x-e^{x}$$
Short Answer
Expert verified
The graph of the function \( y = 4x - e^x \) decreases as \( x \) increases, has a y-intercept at (0, -1), and steepens for larger \( x \) values.
Step by step solution
01
Identify Key Features
First, we need to understand the function \( y = 4x - e^x \). Determine its key characteristics which involve finding any intercepts and asymptotes.
02
Find the Y-intercept
Set \( x = 0 \) to find the y-intercept: \( y = 4(0) - e^0 = -1 \). Thus, the y-intercept is \( (0, -1) \).
03
Find the X-intercept
To find the x-intercepts, set \( y = 0 \): \[ 0 = 4x - e^x \]This equation cannot be solved analytically, but using numerical methods or a calculator could provide an approximate solution. Consider using a graphing calculator to find where the graph crosses the x-axis.
04
Determine End Behavior
Examine the limits as \( x \to \infty \) and \( x \to -infty \):- As \( x \to -\infty \), since the polynomial term \( 4x \) dominates, \( y \to -\infty \).- As \( x \to \infty \), the exponential term \( e^x \) grows much faster than \( 4x \), so \( y \to -\infty \).
05
Analyze the Function's Slope
Find the derivative to understand the slope behavior: \( y' = 4 - e^x \). This function provides insight into where the function is increasing or decreasing:- Since \( e^x \) is always positive, \( y' = 4 - e^x \) will be positive for smaller values of \( x \) and negative for larger values of \( x \).
06
Sketch the Graph
Using the y-intercept, the estimated x-intercept(s), end behavior, and slope analysis, sketch the function. Plot the function on a graph, starting from the y-intercept and tracking how the slope changes the direction of the function curve.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Y-intercept
The Y-intercept of a function is the point where the graph crosses the y-axis. This happens when the value of the variable \( x \) is zero. To find the Y-intercept of the function \( y = 4x - e^x \), we substitute \( x = 0 \). The equation simplifies to:
- \( y = 4(0) - e^0 \)
- Since \( e^0 = 1 \), we have \( y = -1 \)
X-intercept
The X-intercept is the point where the graph crosses the x-axis, which occurs when \( y = 0 \). For the function \( y = 4x - e^x \), we solve:
A calculator can provide a valid visualization to see where the function crosses the x-axis, delivering a practical approach for determining the intercept.
- \( 0 = 4x - e^x \)
A calculator can provide a valid visualization to see where the function crosses the x-axis, delivering a practical approach for determining the intercept.
End Behavior
The end behavior of a function describes what happens to the value of \( y \) as \( x \) approaches positive or negative infinity.
- As \( x \to -\infty \): The term \( 4x \) dominates because it is linear, leading \( y \to -\infty \).
- As \( x \to \infty \): The exponential term \( e^x \) grows much faster than the linear term, causing \( y \to -\infty \) due to the subtraction \( -e^x \).
Derivative
The derivative of a function gives insight into the function's slope or rate of change for different values of \( x \). For \( y = 4x - e^x \), the derivative is:
- When \( e^x < 4 \): \( y' \) is positive, indicating the function is increasing.
- When \( e^x > 4 \): \( y' \) is negative, indicating the function is decreasing.
Since \( e^x \) grows very rapidly, after a certain point the function will start decreasing, explaining why the original function eventually heads towards negative infinity.
- \( y' = 4 - e^x \)
- When \( e^x < 4 \): \( y' \) is positive, indicating the function is increasing.
- When \( e^x > 4 \): \( y' \) is negative, indicating the function is decreasing.
Since \( e^x \) grows very rapidly, after a certain point the function will start decreasing, explaining why the original function eventually heads towards negative infinity.
Exponential Function
An exponential function contains a variable as an exponent. In this function, \( e^x \) is an exponential function where the base is the constant \( e \), approximately 2.718.
In the given function, \( e^x \) influences the increasing and decreasing sections of the graph, where it ultimately guides the end behavior because of its overpowering growth rate.
- Exponential functions grow rapidly; for positive \( x \), they increase quickly.
- In this function, \( e^x \) is subtracted, which significantly impacts the function's shape, as it surpasses linear growth.
In the given function, \( e^x \) influences the increasing and decreasing sections of the graph, where it ultimately guides the end behavior because of its overpowering growth rate.
