Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 2
Graph each equation. $$f(x)=4^{x}$$
Short Answer
Expert verified
Graph an increasing curve passing through (0,1) with a horizontal asymptote at y=0.
Step by step solution
01
Identify the Type of Function
Recognize that the function given, \( f(x) = 4^x \), is an exponential function. In this function, the base is 4 and the exponent is \( x \).
02
Determine Key Characteristics
Identify key characteristics of the exponential function: the domain is all real numbers, the range is \((0, +\infty)\), and the y-intercept is at the point \( (0,1) \). The graph is continuously increasing as the base \(4\) is greater than 1.
03
Plot Key Points
Calculate key points by substituting a few values for \(x\). For example: \(f(-1) = 4^{-1} = \frac{1}{4}\), \(f(0) = 4^0 = 1\), \(f(1) = 4^1 = 4\), and \(f(2) = 4^2 = 16\). Use these points to guide your graph plotting.
04
Sketch the Graph
Based on the calculated points, plot them on a coordinate plane. The points should reflect an exponential curve starting from low values on the y-axis (at and near negative x-values) to high y-values as x increases. The graph should pass through the points (0,1), (1,4), (2,16), and continue rising steeply.
05
Draw Asymptote
Identify the horizontal asymptote of the function. Since \( f(x) = 4^x \) approaches 0 as \( x \to -\infty \), draw a horizontal asymptote along the x-axis, at \( y = 0 \).
06
Review Graph Properties
Review the curve to ensure it is a smooth, continuous line that goes from left to right, approaching the horizontal asymptote at \( y = 0 \) and rising sharply without bound as \( x \) increases. Make sure the graph aligns with the key characteristics identified earlier.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing
Graphing an exponential function like \( f(x) = 4^x \) involves understanding its general curve structure on a coordinate plane. The function increases rapidly as \( x \) increases because the base 4 is greater than 1. To accurately graph this function, it helps to plot some key points. For instance:
Put these points on the graph. They should create a curve that begins gently and then climbs quickly as \( x \) becomes larger. These points are guiding markers to help sketch the characteristic exponential shape, rising more steeply for increasing \( x \) values.
- When \( x = -1 \), \( f(x) = \frac{1}{4} \).
- When \( x = 0 \), \( f(x) = 1 \) (as the value of any number raised to the zero power is 1).
- When \( x = 1 \), \( f(x) = 4 \).
- At \( x = 2 \), \( f(x) = 16 \).
Put these points on the graph. They should create a curve that begins gently and then climbs quickly as \( x \) becomes larger. These points are guiding markers to help sketch the characteristic exponential shape, rising more steeply for increasing \( x \) values.
Function Characteristics
The characteristics of an exponential function such as \( f(x) = 4^x \) are distinct and unique. The domain of an exponential function is all real numbers, meaning \( x \) can take any value from negative infinity to positive infinity. However, the range is different. For exponential growth functions, the range is \((0, +\infty)\), which means all output values of the function are positive.
An essential characteristic is that this function has a y-intercept at \( (0, 1) \), indicating the point where the graph crosses the y-axis. Since the base 4 is greater than 1, the function is increasing, meaning as \( x \) gets larger, \( f(x) \) also becomes larger without bound. These properties make exponential functions powerful for modeling growth processes, compounding phenomena, and other similar real-world situations.
An essential characteristic is that this function has a y-intercept at \( (0, 1) \), indicating the point where the graph crosses the y-axis. Since the base 4 is greater than 1, the function is increasing, meaning as \( x \) gets larger, \( f(x) \) also becomes larger without bound. These properties make exponential functions powerful for modeling growth processes, compounding phenomena, and other similar real-world situations.
Asymptotes
In the context of exponential functions, an asymptote is a line that the graph of the function approaches but never actually touches. For the function \( f(x) = 4^x \), the graph has a horizontal asymptote. This is a line located at \( y = 0 \), or the x-axis. As \( x \) tends to negative infinity, the value of \( 4^x \) approaches zero but never reaches it.
Identifying the asymptote is vital because it shows the boundary behavior of the graph as \( x \) decreases. Essentially, no matter how much \( x \) decreases, \( f(x) \) remains positive and approaches, but does not reach, zero. Draw this asymptote on your graph as a dashed line along the x-axis. Observing the asymptote helps in predicting and understanding the long-term behavior of the function.
Identifying the asymptote is vital because it shows the boundary behavior of the graph as \( x \) decreases. Essentially, no matter how much \( x \) decreases, \( f(x) \) remains positive and approaches, but does not reach, zero. Draw this asymptote on your graph as a dashed line along the x-axis. Observing the asymptote helps in predicting and understanding the long-term behavior of the function.
