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Chapter 4: Problem 8

Graph each function. If you are using a graphing calculator, make a hand-drawn sketch from the screen. $$ y=\left(\frac{1}{5}\right)^{x} $$

Short Answer

Expert verified
Graph is a decreasing exponential function, asymptotic to \( y = 0 \), passing through \((0, 1)\).

Step by step solution

01

Understanding the Function

The function given is an exponential function, which is generally expressed in the form \( y = a^x \). In this case, \( a = \frac{1}{5} \), indicating the base is a fraction less than 1, which makes it a decreasing exponential function.
02

Setting Up a Table of Values

Choose several values for \( x \) to calculate corresponding \( y \) values. Common choices might include \(-2, -1, 0, 1, \) and \( 2 \).- When \( x = -2 \), \( y = \left( \frac{1}{5} \right)^{-2} = 25 \)- When \( x = -1 \), \( y = \left( \frac{1}{5} \right)^{-1} = 5 \)- When \( x = 0 \), \( y = \left( \frac{1}{5} \right)^0 = 1 \)- When \( x = 1 \), \( y = \left( \frac{1}{5} \right)^1 = \frac{1}{5} \)- When \( x = 2 \), \( y = \left( \frac{1}{5} \right)^2 = \frac{1}{25} \)
03

Plotting Points on Graph

Use the table from Step 2 to plot the points \((-2, 25), (-1, 5), (0, 1), (1, \frac{1}{5}),\) and \((2, \frac{1}{25})\) on a coordinate plane. These points will help to form the shape of the exponential curve.
04

Drawing the Exponential Curve

Connect the plotted points with a smooth curve. Since the function is \( y = \left( \frac{1}{5} \right)^x \), the curve should start at a high value when \( x \) is negative and approach zero as \( x \) increases. Remember to draw the curve asymptotic to the x-axis as \( x \to \infty \).
05

Analyzing the Graph's Characteristics

The graph of \( y = \left( \frac{1}{5} \right)^x \) is a decreasing exponential function. It has a horizontal asymptote at \( y = 0 \), passes through the point (0, 1), and decreases as \( x \) increases.

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Key Concepts

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

Decreasing Exponential
Exponential functions are powerful mathematical tools that can model growth or decay. A decreasing exponential function occurs when the base of the exponential, denoted as \( a \), is a fraction between 0 and 1. In the exercise, the function \( y = \left( \frac{1}{5} \right)^x \) is a classic example. Here, the base \( \frac{1}{5} \) tells us everything we need to know about its behavior.
  • Decreasing Pattern: When the base is less than 1, the function decreases as \( x \) increases. This is because each power of the base results in a smaller value.
  • Real-World Applications: These functions can represent processes such as radioactive decay or depreciation in value over time.
  • Key Characteristics: This decay means that for every increase in \( x \), the \( y \)-value gets closer to zero but never actually reaches it.
Graphing Exponential Functions
Graphing exponential functions involves plotting a curve based on calculated \( x \)-\( y \) pairs. This process helps visualize the function's behavior and identify its characteristics clearly.
  • Creating a Table of Values: By choosing different values for \( x \), we calculate the corresponding \( y \)-values. For \( x = -2, -1, 0, 1, 2 \), the respective \( y \)-values for our function are 25, 5, 1, \( \frac{1}{5} \), and \( \frac{1}{25} \).
  • Plotting Points: Each pair forms a coordinate point, such as \((-2, 25)\), to be plotted on a graph. This forms a sequence that defines the shape of the exponential curve.
  • Drawing the Curve: Connect the points smoothly to illustrate the exponential decline from a steep start to a gradual approach to zero.
  • Seamless Curve: The plotted points should connect into a smooth, continuously decreasing curve, depicting the exponential decay accurately.
Horizontal Asymptote
A horizontal asymptote is a line that the curve of a function approaches but never quite touches as \( x \) moves to positive or negative infinity. This concept is crucial for understanding the long-term behavior of exponential functions.
  • Definition: For a decreasing exponential function like \( y = \left( \frac{1}{5} \right)^x \), the horizontal asymptote is at \( y = 0 \).
  • Real-World Analogy: Imagine moving closer and closer to a finishing line but never quite stepping on it鈥攖his is how the graph behaves as it nears the horizontal axis.
  • Importance: The asymptote indicates that no matter how high \( x \) gets, \( y \) will never actually become zero. It will just continue getting smaller.
  • Graphing Insight: When you graph, visually emphasize the graph leveling off as it approaches the asymptote at \( y=0 \).

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Most popular questions from this chapter

The population (in millions) of a city \(t\) years from now is given by the indicated function. a. Find the relative rate of change of the population 8 years from now. b. Will the relative rate of change ever reach \(1.5 \%\) ? $$ P(t)=4+1.3 e^{0.04 t} $$

The manager of a city bus line estimates the demand function to be \(D(p)=150,000 \sqrt{1.75-p}\), where \(p\) is the fare in dollars. The bus line currently charges a fare of $$\$ 1.25$$, and it plans to raise the fare to increase its revenues. Will this strategy succeed?

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=4000 e^{-0.01 p}, \quad p=200 $$

Use your graphing calculator to graph each function on the indicated interval, and give the coordinates of all relative extreme points and inflection points (rounded to two decimal places). [Hint: Use NDERIV once or twice together with ZERO.] (Answers may vary depending on the graphing window chosen.) $$ f(x)=\frac{x^{2}}{e^{x}} \text { for }-1 \leq x \leq 8 $$

For each demand function \(D(p)\) : a. Find the elasticity of demand \(E(p)\). b. Determine whether the demand is elastic, inelastic, or unit-elastic at the given price \(p\). $$ D(p)=\frac{600}{p^{3}}, \quad p=25 $$
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