Question

Graph the function and its parent function. Then describe the transformation. \(f(x)=-2\)

Short answer

The transformation is a vertical down shift by 2 units from the parent function \(f(x) = 0\) to the function \(f(x) = -2\).

Step-by-step solution

Graph of the parent function

Let's begin by first drawing the parent function \(f(x) = 0\). This is a straight horizontal line that crosses the y-axis at y=0.

Graph of our function

Now, let's graph our function \(f(x) = -2\). This is also a straight horizontal line but it crosses the y-axis at y=-2.

Description of the transformation

Comparing the graph of our function with the parent function, we can see that there is a vertical shift downwards by 2 units. The line has moved from y=0 to y=-2. This is because the function \(f(x)\) is a constant equal to -2.

Key concepts

Parent Function

A parent function is like the blueprint of a family of functions. It is the simplest form of a function type from which other functions can be derived through transformations.
For linear functions, the parent function is usually given as \( f(x) = 0 \), which is a horizontal line crossing the y-axis at y = 0. This represents a graph with no slope or curvature changes.
Understanding the parent function is crucial because it provides a reference point to see how other functions differ through transformations.

Vertical Shift

A vertical shift involves moving a graph up or down on the coordinate plane without changing its shape or orientation. When we add or subtract a constant from a function, the graph of the function moves vertically.
In our exercise, the function transitions from the parent function \( f(x) = 0 \) to \( f(x) = -2 \). This is a vertical shift downward by 2 units, as all points on the function move directly down.
A positive shift moves the graph up, while a negative shift moves it down. These changes help adjust the function's position to suit particular needs or data sets.

Constant Function

A constant function is a type of function where the output value is the same for any input value. It can be represented by \( f(x) = c \), where \( c \) is a constant value.
The graph of a constant function is a horizontal line at y = c. In our case, the function \( f(x) = -2 \) means every input results in an output of -2, forming a straight line at y = -2.
Constant functions are easy to graph and understand, making them a foundational concept in mathematics. They help illustrate simple transformations like vertical shifts and offer a clear example of functions unaffected by the variable \( x \).