Problem 22 Describe the transformation of \... [FREE SOLUTION]

91影视

BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 6: Problem 22

Describe the transformation of $f$ represented by $g$. Then graph each function. $f(x)=e^{-x}, g(x)=e^{-5 x}+2$

Short Answer

Expert verified

The function $g(x) = e^{-5x} + 2$ is a transformation of the function $f(x) = e^{-x}$ that represents a faster decay rate (5 times faster) and a vertical shift upwards by 2 units.

Step by step solution

Identification

Identify the original function and its transformation. Here, the base function is $f(x) = e^{-x}$. The transformed function is $g(x) = e^{-5x} + 2$.

Determine the Rate of Exponential Decay

Compare the two functions to find the rate of exponential decay. The rate of decay in $f(x) = e^{-x}$ is 1, while the decay rate has increased five times in $g(x) = e^{-5x} + 2$. This means the decay in $g(x)$ is faster than in $f(x)$.

Determine the Vertical Shift

We notice a transformation of $+2$ in the $g(x)$ function, which implies there is a vertical shift of 2 units upwards. Hence the graph of $g(x)$ is the graph of $f(x)$ shifted two units upwards.

Graph the Functions

Now, graph the functions in a coordinate plane. The graph of $f(x) = e^{-x}$ is a downward-sloping curve, starting at $y = 1$ when $x = 0$ and asymptotically approaching $y = 0$ as $x$ increases. The graph of $g(x) = e^{-5x} + 2$ is the same downward-sloping curve shifted up 2 units, starting at $y = 3$ when $x = 0$ and asymptotically approaching $y = 2$ as $x$ increases, decaying five times faster than $f(x)$.

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.

Function Transformation

In mathematical terms, function transformation involves altering a given function in specific ways to produce new functions. It's like applying a set of rules to shift, stretch, or contract a graph around a plane.

For the given functions, the transformation from $ f(x) = e^{-x} $ to $ g(x) = e^{-5x} + 2 $ involves two key changes:

Stretching/Shrinking: The exponential decay factor changes from $ -x $ to $ -5x $. This implies a horizontal compression, making the function decay faster.
Vertical Shift: An addition of $+2$ shifts the entire graph upwards. It affects where the curve approaches zero, changing asymptotic behavior.

These transformations allow us to clearly see how changes in equations affect the graph's shape and position.

Exponential Decay

Exponential decay describes a process where quantities are reduced at a consistent rate over time. It鈥檚 common in natural processes like radioactive decay or cooling of hot objects.

In the function $ f(x) = e^{-x} $, the decay rate is consistent such that it decreases gradually.

When you look at $ g(x) = e^{-5x} + 2 $, the decay occurs five times faster due to the factor of 5 in the exponent. Faster exponential decay means the function decreases more rapidly.

Understanding this concept helps in determining how quickly a curve will drop, which can be useful in predicting long-term behaviors or instantaneous changes in phenomena.

Graphing Functions

Graphing functions is a crucial skill to visually understand how functions behave. It involves plotting values on a coordinate system to form a curve or line.

Let's start with $ f(x) = e^{-x} $. This creates a smooth, downward-sloping curve starting from $ y = 1 $ when $ x = 0 $, and it approaches the x-axis as $ x $ increases.

For $ g(x) = e^{-5x} + 2 $, the graph will also slope downwards but much steeper due to the faster decay rate. It shifts the starting point to $ y = 3 $, moving similarly to $ f(x) $ but with a faster approach towards an asymptote at $ y = 2 $.

Practice graphing these functions to master their behaviors and transformations.

Vertical Shift

A vertical shift in functions means moving the entire graph up or down by a certain number of units. This doesn鈥檛 affect the shape of the graph, just its position on the y-axis.

For instance, in transforming $ f(x) = e^{-x} $ to $ g(x) = e^{-5x} + 2 $, the $+2$ indicates a vertical shift. Specifically, it moves the graph 2 units upwards.

This can be visualized by noting the graph of $ g(x) $ starts at $ y = 3 $ instead of $ y = 1 $ like $ f(x) $, and it asymptotically approaches $ y = 2 $ instead of zero.

Vertical shifts help in highlighting constant changes or offsets in functions that could represent real-world adjustments in data or predictions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=e^{-x}, g(x)=e^{-5 x}+2\)

Short Answer

Step by step solution

Identification

Determine the Rate of Exponential Decay

Determine the Vertical Shift

Graph the Functions

Key Concepts

Function Transformation

Exponential Decay

Graphing Functions

Vertical Shift

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.