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Chapter 0: Problem 25

Sketch a graph of the given function. $$f(x)=3 e^{-2 x}$$

Short Answer

Expert verified
The graph of \(f(x)=3e^{-2x}\) shows exponential decay as x increases, and exponential growth as x decreases. The function approaches the line \(y=0\) as a horizontal asymptote.

Step by step solution

01

Identify critical values and behavior

For \(f(x)=3e^{-2x}\), the coefficient '3' is positive which means the function is upward. The negative sign in the exponent \( -2x \) indicates the graph of the function is reflected across the y-axis. Exponential growth occurs as x decreases.
02

Calculate the value of the function at several points

Calculate the value of the function at several points, for example, at \(x=-1, 0, 1\). For \(x=-1\), \(f(x)=3e^{2*1}=3*7.39=22.17\). For \(x=0\), \(f(x)=3e^{-2*0}=3*0=3\). For \(x=1\), \(f(x)=3e^{-2*1}=3*0.368=1.1.
03

Plot the points on a graph

Plot these points (-1, 22.17), (0, 3) and (1, 1.1) on a graph. Connect the points with a smooth curve.
04

Understand the behavior of the function

The graph of the function should show exponential decay as x increases and exponential growth as x decreases. The function approaches \(y=0\) as \(x \to \infty\), which is a horizontal asymptote.
05

Sketch the full graph

By combining all data points together with the curve's behavior, complete sketching the graph of the given function.

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Key Concepts

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

Exponential Decay
Exponential decay describes a process where the quantity decreases rapidly at first and then slowly over time. In the context of the function provided, \( f(x) = 3e^{-2x} \), the exponential decay behavior is determined by the negative exponent.

When you see an exponent with a negative value, such as \(-2x\), it indicates exponential decay. The "\( e \)" in the function represents Euler's number, which is approximately 2.718. Here are some key points to understand this concept better:
  • The rate of decay is controlled by the exponent. A more negative value results in a steeper curve, indicating quicker decay.
  • The base of the exponential function, which is \( e \), remains constant, and the coefficient "3" affects the initial value when \( x = 0 \).
  • As \( x \) increases, the value of \( e^{-2x} \) becomes increasingly smaller, leading \( f(x) \) to decrease rapidly at first.
Understanding this behavior is crucial when graphing because it helps predict how the curve will look as you move along the x-axis.
Horizontal Asymptote
A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches as \( x \) tends towards infinity. For the function \( f(x) = 3e^{-2x} \), there is a horizontal asymptote at \( y = 0 \). This is because exponential functions of the form \( e^{kx} \) approach zero as \( x \) gets very large when \( k \) is negative.

Key characteristics of the horizontal asymptote in this function:
  • As \( x \to \infty \), the value of \( e^{-2x} \) approaches zero, making the entire function \( f(x) \) approach zero as well.
  • The horizontal asymptote does not limit how low the function can go at extreme negative \( x \) because it specifically refers to behavior at large positive \( x \).
  • This asymptotic behavior helps in predicting the end behavior of the graph, indicating the function will flatten out as it moves right along the x-axis.
The notion of a horizontal asymptote provides insight into the ultimate trend of the function, facilitating accurate graph sketching.
Graph Sketching
Graph sketching is a visual representation of a mathematical function, focusing on its key characteristics such as shape, intercepts, and asymptotes. When sketching the graph of \( f(x) = 3e^{-2x} \), it is important to use the information we know about exponential decay and the horizontal asymptote to draw a precise curve.

To effectively sketch this function's graph, follow these steps:
  • Plot Key Points: Calculate and plot key points such as \((-1, 22.17)\), \((0, 3)\), and \((1, 1.1)\). These values were derived by substituting \( x \) values into the function and computing \( f(x) \).
  • Connect the Points: Connect these points smoothly to reflect the exponential decay, making sure the curve flattens as it moves right to reflect the horizontal asymptote at \( y = 0 \).
  • Indicate Asymptotic Behavior: Mark the horizontal asymptote on the graph to show that, as \( x \) increases positively, the graph gets closer to \( y = 0 \) without ever crossing it.
Practicing graph sketching with this approach, relying on analysis and theoretical insights, builds a stronger understanding of the interplay between algebraic expressions and their graphic interpretations.

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Most popular questions from this chapter

A fast-food restaurant gives every customer a game ticket. With each ticket, the customer has a 1 -in- 10 chance of winning a free meal. If you go 10 times, estimate your chances of winning at least one free meal. The exact probability is \(1-\left(\frac{9}{10}\right)^{10} .\) Compute this number and compare it to your guess.

Find all vertical asymptotes. $$f(x)=\frac{4 x}{x^{2}+4}$$

In golf, the goal is to hit a ball into a hole of diameter 4.5 inches. Suppose a golfer stands \(x\) feet from the hole trying to putt the ball into the hole. A first approximation of the margin of error in a putt is to measure the angle \(A\) formed by the ray from the ball to the right edge of the hole and the ray from the ball to the left edge of the hole. Find \(A\) as a function of \(x .\)

For \(y=x^{3},\) describe how the graph to the left of the \(y\) -axis compares to the graph to the right of the \(y\) -axis. Show that for \(f(x)=x^{3},\) we have \(f(-x)=-f(x),\) In general, if you have the graph of \(y=f(x)\) to the right of the \(y\) -axis and \(f(-x)=-f(x)\) for all \(x,\) describe how to graph \(y=f(x)\) to the left of the \(y\) -axis.

Sketch a graph of the function showing all extreme, intercepts and asymptotes. $$f(x)=\frac{x-1}{x^{2}+4 x+3}$$
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