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Chapter 3: Problem 14

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). $$ y=f(x)+5 $$

Short Answer

Expert verified
The graph of \( y=f(x)+5 \) is a vertical shift of \( f(x) \) upwards by 5 units.

Step by step solution

01

Understand the Original Function

The given transformation is based on the original function, which is defined as \( y = f(x) \). This graph is the one we will compare the transformation to.
02

Identify the Transformation Type

The function transformation given is \( y = f(x) + 5 \). This represents a vertical translation or shift of the graph of the function.
03

Determine the Direction and Magnitude of Shift

The expression \( +5 \) indicates that the graph of \( f(x) \) is being shifted vertically upward by 5 units. The entire graph, including all points on the original curve, moves up.
04

Describe the Transformed Graph

After applying the transformation \( y = f(x) + 5 \), every point on the graph of the original function \( f(x) \) moves 5 units up along the y-axis. Relative positions between points on the graph remain unchanged.

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Key Concepts

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

Vertical Translation
A vertical translation is a type of function transformation that shifts a graph up or down along the y-axis. This happens when you add or subtract a constant from the function's output. In the original exercise, the function transformation given is \( y = f(x) + 5 \). Here, the entire graph of the original function is moved upward by 5 units. This type of transformation does not affect the x-coordinates of the graph's points. Instead, it increases all y-coordinates by the same value.
  • A positive constant will shift the graph upwards.
  • A negative constant will shift the graph downwards.
This transformation is simple yet powerful, allowing you to change the position of the graph without altering its shape or orientation.
Graph of a Function
The graph of a function is a visual representation of the relationships between the input (x) and output (y) of a function. Each point on the graph \(x, y\) satisfies the function equation such that \(y = f(x)\). Regardless of how a function is represented algebraically, its graph shows how y-values change as x-values change.
  • The x-axis represents the input values or domain.
  • The y-axis represents the output values or range.
Understanding the graph helps in recognizing patterns and summarizing key characteristics such as slope and intercepts. A transformation does not change the domain or range specifics but can alter their physical location on the coordinate plane.
Algebraic Transformation
Algebraic transformation refers to the modification of a function's equation to obtain a new, transformed function. This involves operations like addition, subtraction, multiplication, or division by constants. In our example, the transformation is presented as \(y = f(x) + 5\), where we add a constant 5 to the original function \(f(x)\).
  • Addition or subtraction results in vertical translations.
  • Addition can also suggest an adjustment in the function's intercepts.
Algebraic transformations are essential for manipulating and understanding the effects of changes within functions, allowing a shift in perspective without changing the overall behavior.
Shift of Graph
A shift of graph, particularly in the context of function transformations, refers to the entire movement of a graph from one location on the coordinate plane to another. This movement happens without changing the graph's shape or orientation. In the case of the vertical translation \(y = f(x) + 5\), the shift is linear and direct.
The new graph is exactly five units above its original position. This maintains consistent spacing between all points in the graph.
  • Horizontal shifts involve changes to the x-coordinates by transforming the input as \(f(x\pm h)\).
  • Vertical shifts, like in our example, occur by modifying the output as \(f(x) \pm k\).
Understanding shifts helps predict how changes to one part of a function's equation influence the graph as a whole.

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Most popular questions from this chapter

For the following exercises, let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1\). True or False: \((f \circ g)(x)=F(x)\).

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The number of bacteria in a refrigerated food product is given by $$ N(T)=23 T^{2}-56 T+1,3

For the following exercises, find \(f^{-1}(x)\) for each function. $$ f(x)=3-x $$
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