Problem 19 - Algebra 2

Question

Graph each function. \(y=\sqrt[3]{x-4}\)

Step-by-Step Solution

Answer

Firstly, understand the general shape of the cube root function. Then, observe that the 'minus 4' within the cube root function shifts the graph 4 units to the right. The origin of the new graph will be at (4,0) instead of (0,0). The curve goes off to positive infinity for values greater than 4, and to negative infinity for values less than 4. The graph of the function \(y=\sqrt[3]{x-4}\) will maintain the usual shape of a cube root function, which is symmetrical about the origin.

Step 1: Understanding the Cube Root Function

First, one needs to understand how the cube root function behaves. A cube root function \(y = \sqrt[3]{x}\) is a radical function whose curve starts at negative infinity, passes through the origin (0,0), and goes through to positive infinity. The cube root function is symmetric with respect to the origin, meaning it's unchanged if \(x\) and \(y\) are both replaced by their negatives. Thus, a positive input gives a positive output and a negative input gives a negative output.

Step 2: Shift of the Graph

In our given function \(y=\sqrt[3]{x-4}\), the number 4 inside the cube root is a horizontal shift of the graph. Here, 4 units are being subtracted from \(x\). The graph of the function will be shifted 4 units to the right compared to the graph of the basic cube root function \(y=\sqrt[3]{x}\). Therefore, our curve will start at (4,0) instead of (0,0).

Step 3: Graph the Function

We start the graph at (4,0) instead of (0,0) due to the 4 units shift to the right. The curve goes off to positive infinity for values greater than 4, and to negative infinity for values less than 4. The shape of the curve will follow the typical behavior of cube root function which is symmetrical about the point (4,0).

Key Concepts

Graphing Functions

Graphing functions is an essential skill in mathematics. When you graph functions, you're creating a visual representation of a mathematical relationship, helping to clarify how values behave and relate to each other. For the cube root function, specifically, its graph spans all real numbers. The basic form, \( y = \sqrt[3]{x} \), has a curve that is smooth and continuous, unlike the square root function which only includes non-negative values of \( x \).
Here are some features of graphing a cube root function:

• The graph passes through the origin (0,0), serving as a central reference point.
• As \( x \) approaches negative infinity, \( y \) decreases infinitely. As \( x \) approaches positive infinity, \( y \) increases indefinitely.
• This behavior shows a balanced growth on both sides of the origin.
• The graph is not a line; it is a curve which is more gradual near the origin.

Graphing this function often involves plotting key points and recognizing its continuous nature.

Horizontal Shift

A horizontal shift changes where the function begins on the x-axis without changing its shape. This is seen in the function \( y=\sqrt[3]{x-4} \), where the graph shifts horizontally by subtracting 4 from \( x \).
Understanding how this affects the graph:

• This subtraction means the standard graph will move 4 units to the right.
• While \( y = \sqrt[3]{x} \) starts at (0,0), the shifted function \( y = \sqrt[3]{x-4} \) will start at (4,0).
• The whole graph, all other points included, moves right; the shape and direction remain intact.

Horizontal shifts are crucial for understanding more complex transformations and can simplify how we visualize changes to the graph.

Symmetry in Functions

Symmetry in functions reveals balanced qualities in their graphs. The cubic root function \( y=\sqrt[3]{x} \) is a prime example of describing symmetry across the origin (0,0). This means both halves of the graph mirror each other around this central point. Every input (\( x \)) that has a corresponding output (\( y \)) also has a counterpart \((-x, -y)\) with equivalent magnitude but opposite sign.
Key points to note about symmetry:

• The origin is the central point for cubic root functions' symmetry.
• This symmetry is also known as rotational symmetry of order 2, showcasing how the function could rotate 180 degrees around the origin and look the same.
• Symmetry helps simplify the process of graphing and analyzing functions because knowing one part of the graph instantly gives information about another part.

In transformations like \( y=\sqrt[3]{x-4} \), the symmetry point becomes (4,0) due to the shift, but the property remains fundamentally the same.