To determine the graph of the function [tex]\( y = \sqrt[3]{x-5} \)[/tex], it's crucial to understand the basic characteristics and transformations involved. Let's break down the function step by step:
### Step 1: Understanding the Cubic Root Function
The basic cubic root function is [tex]\( y = \sqrt[3]{x} \)[/tex]. This function has the following characteristics:
- It is an odd function, meaning it is symmetric about the origin.
- The graph passes through the origin (0, 0).
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases, and as [tex]\( x \)[/tex] decreases, [tex]\( y \)[/tex] decreases.
- The graph extends infinitely in all directions.
### Step 2: Horizontal Shift
The given function is [tex]\( y = \sqrt[3]{x - 5} \)[/tex]. This is a horizontal shift of the basic cubic root function:
- The term [tex]\( x - 5 \)[/tex] indicates a shift to the right by 5 units.
### Step 3: Key Points and Plotting
To plot the function, consider a few key points:
1. When [tex]\( x = 5 \)[/tex]:
[tex]\[
y = \sqrt[3]{5 - 5} = \sqrt[3]{0} = 0
\][/tex]
The point (5, 0) lies on the graph.
2. When [tex]\( x = 6 \)[/tex]:
[tex]\[
y = \sqrt[3]{6 - 5} = \sqrt[3]{1} = 1
\][/tex]
The point (6, 1) lies on the graph.
3. When [tex]\( x = 4 \)[/tex]:
[tex]\[
y = \sqrt[3]{4 - 5} = \sqrt[3]{-1} = -1
\][/tex]
The point (4, -1) lies on the graph.
4. When [tex]\( x = 13 \)[/tex]:
[tex]\[
y = \sqrt[3]{13 - 5} = \sqrt[3]{8} = 2
\][/tex]
The point (13, 2) lies on the graph.
5. When [tex]\( x = -3 \)[/tex]:
[tex]\[
y = \sqrt[3]{-3 - 5} = \sqrt[3]{-8} = -2
\][/tex]
The point (-3, -2) lies on the graph.
### Summary of the Graph Characteristics
- The graph of [tex]\( y = \sqrt[3]{x - 5} \)[/tex] is the same shape as [tex]\( y = \sqrt[3]{x} \)[/tex], but shifted 5 units to the right.
- The curve crosses the x-axis at (5, 0).
- For [tex]\( x > 5 \)[/tex], the function values (y) are positive and increase.
- For [tex]\( x < 5 \)[/tex], the function values (y) are negative and decrease.
To identify the correct choice from given graphs, look for one with these characteristics, particularly the horizontal shift to the right.
