Problem 16

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function $f$. $$ y=f(x)-2 $$

Step-by-Step Solution

Verified

Answer

The graph is shifted downward by 2 units.

1Step 1: Identify the transformation applied

The expression given is $y = f(x) - 2$. This represents a transformation on the original function $f(x)$. To understand what kind of transformation this is, note that the $-2$ is subtracted from the whole function $f(x)$. This indicates a vertical transformation.

2Step 2: Describe the vertical shift

A transformation of the form $y = f(x) - c$ where $c$ is a positive constant, represents a vertical shift downward by $c$ units. Here, $c = 2$, so this transformation shifts the graph of $f(x)$ downward by 2 units.

3Step 3: Sketch the transformation

To help visualize the transformation, imagine taking every point on the graph of $f(x)$ and moving it two units down along the y-axis. This results in the entire graph being lowered by 2 units, while its shape and other properties (like width, x-intercepts, etc.) remain unchanged.

Key Concepts

Vertical ShiftGraph TransformationFunction Graph

Vertical Shift

A vertical shift in a function graph is one of the simplest types of graph transformations. It involves moving the entire graph of a function up or down along the y-axis. When you see an expression like $y = f(x) - c$, this means the function is being shifted vertically.- If $c$ is positive, the graph shifts downward.- If $c$ is negative, the graph shifts upward.For example, in the given function $y = f(x) - 2$, we are looking at a vertical shift of 2 units downward. This results because the "-2" signifies that every point on the graph of the original function $f(x)$ will move down exactly 2 units.

Graph Transformation

Graph transformation involves changing the position, size, or orientation of a function's graph. This can include shifts, stretches, compressions, and reflections. Transformations either modify the graph's appearance or move it within the coordinate plane, without altering the core properties of the function. In our example, we focus on a vertical shift, which is a type of graph transformation. Other transformations can include:

Horizontal shifts: Moving left or right along the x-axis.
Vertical stretches or compressions: Changing the steepness or flatness by multiplying the function.
Reflections: Flipping the graph over a specific axis.

Understanding these transformations helps in easily predicting what a function's graph will look like after changes.

Function Graph

The graph of a function is a visual representation of the mathematical equation. It provides insight into the behavior of the function, showcasing key features like intercepts, maxima, and minima.When graphing a function, even a small transformation like a vertical shift can change your perspective on how the function behaves. In the case of $y = f(x) - 2$:

Each point on the function $f(x)$ moves down two units.
The x-coordinates of the points remain unchanged, only the y-values decrease by 2.

Seeing the graph provides a clearer picture of these shifts, making it a crucial tool in understanding transformations.

Problem 16

Other exercises in this chapter

Problem 15

For the following exercises, determine whether the relation represents $y$ as a function of $x$. $$ x=\frac{3 y+5}{7 y-1} $$

View solution

Problem 16

Given $f(x)=\frac{x}{2+x}$ and $g(x)=\frac{2 x}{1-x}:$ a. Find $f(g(x))$ and $g(f(x))$. b. What does the answer tell us about the relationship between \

View solution

Problem 16

For the following exercises, find a domain on which each function $f$ is one- to-one and non decreasing. Write the domain in interval notation. Then find the

View solution

Problem 16

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph. $$ y=|x-1| $$

View solution