Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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.