Question

Write the function whose graph is the graph of \(y=x^{3},\) but is: Shifted down 4 units

Short answer

y = x^{3} - 4

Step-by-step solution

Understand the original function

The original function given is the cubic function: y = x^{3} Imagine this graph in the coordinate plane.

Identify the transformation

The problem states that the function needs to be shifted down by 4 units. Vertical shifts are adjustments to the y-values.

Apply the vertical shift

To shift the graph of a function down by 4 units, subtract 4 from the original function. This means the new function will be: y = x^{3} - 4

Key concepts

Cubic Functions

A cubic function is a function of the form \(y = ax^{3}+bx^{2}+cx+d\) where \(a, b, c,\) and \(d\) are constants. The simplest cubic function is \(y = x^{3}\). This function has a characteristic 'S' shaped curve which passes through the origin (0,0). A cubic function can have one or three real roots and is capable of changing direction twice.
Key features of cubic functions include:

• They are symmetric about the origin.
• As \(x\) goes to positive or negative infinity, \(y\) also goes to positive or negative infinity respectively.
• Cubic functions have a point of inflection where the curve changes concavity.

Understanding cubic functions is crucial as they form the basis for more complex polynomial functions.

Graph Transformations

Graph transformations involve altering the appearance of the graph of a function without changing its basic shape. Common transformations include:

• Vertical Shifts: Moving the graph up or down.
• Horizontal Shifts: Moving the graph left or right.
• Reflections: Flipping the graph over an axis.
• Stretches and Compressions: Changing the graph's steepness or width.

To perform a vertical shift, you add or subtract a constant from the function. In the example provided, you are asked to shift the cubic function \(y = x^{3}\) down by 4 units. To do this, simply subtract 4 from the function: \(y = x^{3} - 4\). This adjustment affects all the y-values of the function equally, resulting in the entire graph moving downward.

Function Notation

Function notation is a way of expressing functions in a clear and concise manner. The general form of function notation is \(f(x) = \text{expression}\), where \(f\) denotes the function and \(x\) is the variable. For example, \(f(x) = x^{3}\) indicates that the function \(f\) takes input \(x\) and maps it to \(x^{3}\).
Function notation is helpful because it:

• Makes complex expressions easier to read.
• Clarifies which variable is the input.
• Allows easy substitution of values into the function.

In transforming functions, function notation helps you see how changes to the function's expression affect its graph. Using our example, starting with \(f(x) = x^{3}\) and changing it to \(g(x) = x^{3} - 4\) shows that the graph of \(f\) is being shifted down by 4 units to form \(g\). This clear notation is crucial for understanding and manipulating functions efficiently.