Question

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

Short answer

The transformed function is \(y = (x - 4)^3\).

Step-by-step solution

- Understand the basic function

The given basic function is \(y = x^3\). This is the graph we need to transform.

- Identify the transformation needed

The exercise asks for the graph to be shifted to the right by 4 units.

- Apply the horizontal shift

To shift the graph of a function to the right by \(c\) units, replace \(x\) with \(x - c\) in the function. Here, \(c = 4\).

- Write the transformed function

Replacing \(x\) with \(x - 4\) in the original function \(y = x^3\), the transformed function becomes \(y = (x - 4)^3\).

- Verify the transformation

Double-check the expression to ensure that shifting to the right 4 units was correctly applied. The final function is \(y = (x - 4)^3\).

Key concepts

Cubic Functions

Cubic functions are polynomial functions of degree three. A general cubic function is written as:
\[ f(x) = ax^3 + bx^2 + cx + d \]
Here, the coefficients 'a', 'b', 'c', and 'd' are constants, and 'a' is not equal to zero.
The most basic cubic function is \( y = x^3 \), which is a simple polynomial without any additional terms.
It has a characteristic S-shape, called an inflection point, at the origin (0, 0).

When graphing cubic functions, it is important to note how changes in the coefficients affect the graph.
The leading term \( ax^3 \) will determine the end behavior of the function.
Another critical aspect is how transformations can change the appearance and position of the graph.

Function Transformation

Function transformation involves changing the appearance or position of a graph in a coordinate system.
These transformations can include:

• Translations (shifting the graph vertically or horizontally)
• Reflections (flipping the graph over a line)


• Stretches and Compressions (making the graph taller/skinnier or shorter/wider)

In this particular case, the transformation we are dealing with is a horizontal shift.
To shift a function horizontally, we adjust the x-values in the function's equation.
Specifically, if you want to shift the graph of a function \( f(x) \) to the right by 'c' units, you replace \( x \) with \( x - c \) in the function.

So, for the function \( y = x^3 \) and shifting it to the right by 4 units, we get:
\[ y = (x - 4)^3 \]

Graph Shifting

Graph shifting refers to moving a graph horizontally or vertically on the coordinate plane without changing its shape.

**Horizontal Shifting** involves moving the graph left or right:

• To shift the graph to the right by 'c' units, replace each \( x \) with \( x - c \)
• To shift the graph to the left by 'c' units, replace each \( x \) with \( x + c \)

In our exercise, we shifted the graph of \( y = x^3 \) 4 units to the right, giving us \( y = (x - 4)^3 \).

**Vertical Shifting** involves moving the graph up or down:

• To shift the graph up by 'c' units, add 'c' to the function \( f(x) \), like \( f(x) + c \)
• To shift the graph down by 'c' units, subtract 'c' from the function \( f(x) \), like \( f(x) - c \)

Understanding these shifts can make it easier to predict and draw the graph of a function after transformation. This understanding is crucial while solving problems and checking work.