Question

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

Short answer

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

Step-by-step solution

Understand the Original Function

The original function is given by the equation: \[ y = x^3 \].

Identify the Horizontal Shift

To shift the graph to the left by 4 units, we need to modify the function by adding 4 to the input variable (x) inside the function.

Formulate the New Function

Incorporate the horizontal shift into the original function: \[ y = (x + 4)^3 \].

Check Your Work

Verify the transformation by plugging in a few values of x to ensure the graph has shifted left by 4 units. For example, if the original point (0, 0) is in the function \( y = x^3 \), the new function \( y = (x + 4)^3 \) will have the point (-4, 0).

Key concepts

graph transformations

Graph transformations are changes made to the graph of a function that alter its position, shape, orientation or size. These transformations include translations, reflections, stretches, and compressions. A translation shifts the graph horizontally or vertically. This exercise focuses on horizontal translations.

For example, let's take the basic cubic function: \[ y = x^3 \] A horizontal shift means moving the graph left or right along the x-axis.

• To shift left by 4 units, we replace x with \(x + 4\): \[ y = (x + 4)^3 \]

This transformation doesn't affect the shape of the graph. It merely moves every point to the left by 4 units.

polynomial functions

Polynomial functions are expressions involving variables raised to whole number exponents and their coefficients. They have various forms, but we'll focus on cubic functions in this context. The general form of a polynomial function is:
\[ P(x) = a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0 \] For a cubic function, it simplifies to:
\[ y = ax^3 + bx^2 + cx + d \] Here, the function \(y = x^3\) represents a specific type of polynomial where the leading term has the exponent 3 and the coefficient of 1. Polynomial functions like this exhibit smooth and continuous graphs.
By understanding polynomial functions, we grasp how transformations, such as horizontal shifts, affect their graphs.

function transformation

Function transformations involve altering a function's formula to produce changes in its graph. These transformations include:

• Horizontal shifts
• Vertical shifts
• Reflections
• Stretches and compressions

In your exercise, you're focusing on a horizontal shift. The general rule for a horizontal shift to the left by 'h' units is to replace \(x\) with \(x + h\) in the function.
Applying this to \(y = x^3\), we get:
\[ y = (x + 4)^3 \]
By making this replacement, the whole graph shifts left by 4 units. Each point on the original graph of \(y = x^3 \) moves 4 units to the left, resulting in the graph of \(y = (x+4)^3 \). Understanding function transformations like these helps you modify and predict the behavior of mathematical graphs.