/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Describe how each function is a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Describe how each function is a transformation of the original function \(f(x)\) $$ f(x)+5 $$

Short Answer

Expert verified
The function \( f(x) + 5 \) is a vertical shift of the original function \( f(x) \) by 5 units upwards.

Step by step solution

01

Identify the original function

The original function given is \( f(x) \). This is the function that we will be transforming.
02

Recall vertical transformations

Adding or subtracting a constant from a function represents a vertical transformation. When a constant is added, the graph shifts upwards, and when a constant is subtracted, the graph shifts downwards.
03

Analyze the transformation

The transformation given is \( f(x) + 5 \). Since 5 is being added to \( f(x) \), the graph of the function will shift 5 units upward.

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.

Vertical Transformation in Functions
A vertical transformation involves moving the graph of a function up or down along the y-axis. This type of transformation occurs when you add or subtract a constant from a function's equation. It is often the easiest to spot when given a function formula.

For instance, with the original function represented as \( f(x) \), any form of \( f(x) + c \) or \( f(x) - c \) indicates a vertical transformation.

  • If you add a positive constant \( c \), the graph rises.
  • If you subtract a positive constant, the graph falls the same amount.
This simple alteration in the function's formula adjusts positions without changing the graph's shape, which is crucial to understanding transformations.
Exploring Graph Shifts
Graph shifts are essential visual changes in a function鈥檚 graph due to transformations. In terms of vertical transformations, graph shifts occur entirely along the vertical axis.

If considering the transformation \( f(x) + 5 \), it tells us that every point on the original graph of \( f(x) \) moves 5 units upward without affecting the x-coordinate of the points. Here鈥檚 why:
  • Vertical movement: Only the y-values change, represented by adding 5 to each output value of the function, moving every point vertically up.
  • Unchanged x-coordinates: Since no x-component is involved, the horizontal position doesn鈥檛 alter, maintaining original left-to-right placement of the graph.
Vertical graph shifts help in adjusting the graph within its vertical space while preserving its horizontal context.
Understanding the Original Function
When examining transformations, it's crucial to start with the original function in its initial form. This is typically expressed simply, like \( f(x) \). It's the base function that will undergo the planned transformation effects.

Understanding the original function means knowing its basic shape, key points, and its general behavior (like increases or decreases). For example, if \( f(x) \) were a linear function like \( f(x) = x \), you would see a straight diagonal line through the origin before any transformation.
  • Baseline for comparison: Always returning to it helps in clearly seeing how a transformation alters it.
  • Identifying transformations: Knowing \( f(x) \) lets you easily identify what aspects, like vertical shifts, change the graph.
The original function is the starting reference to track and visualize any transformation effects systematically.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Select all of the following tables which represent \(y\) as a function of \(x\). a. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -2 \\ \hline 3 & 1 \\ \hline 4 & 6 \\ \hline 8 & 9 \\ \hline 3 & 1 \\ \hline \end{array} $$ b. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-1 & -4 \\ \hline 2 & 3 \\ \hline 5 & 4 \\ \hline 8 & 7 \\ \hline 12 & 11 \\ \hline \end{array} $$ c. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -5 \\ \hline 3 & 1 \\ \hline 3 & 4 \\ \hline 9 & 8 \\ \hline 16 & 13 \\ \hline \end{array} $$ d. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-1 & -4 \\ \hline 1 & 2 \\ \hline 4 & 2 \\ \hline 9 & 7 \\ \hline 12 & 13 \\ \hline \end{array} $$

Describe how each function is a transformation of the original function \(f(x)\). $$ 4 f(x) $$

Describe how each function is a transformation of the original function \(f(x)\). $$ -f(3 x) $$

Select all of the following tables which represent \(y\) as a function of \(x\). a. $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 2 & 6 & 13 \\ \hline \boldsymbol{y} & 3 & 10 & 10 \\ \hline \end{array} $$ b. $$ \begin{array}{|l|l|l|l|} \hline x & 2 & 6 & 6 \\ \hline y & 3 & 10 & 14 \\ \hline \end{array} $$ c. $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 2 & 6 & 13 \\ \hline \boldsymbol{y} & 3 & 10 & 14 \\ \hline \end{array} $$

Describe how each function is a transformation of the original function \(f(x)\) $$ f(x+4)-1 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.