To identify the graph that accurately represents the function [tex]\( y = \sqrt[3]{x - 5} \)[/tex], let's go through a detailed step-by-step reasoning, starting with some key points and important characteristics of this function.
### 1. Understanding the Function
The function [tex]\( y = \sqrt[3]{x - 5} \)[/tex] is a cube root function that will produce different [tex]\( y \)[/tex]-values based on the input [tex]\( x \)[/tex]-values. Its general behavior can be understood from its definition:
- The inside of the cube root, [tex]\( x - 5 \)[/tex], determines the horizontal shift of the basic cube root function [tex]\( y = \sqrt[3]{x} \)[/tex].
- The function [tex]\( y = \sqrt[3]{x} \)[/tex] is shifted to the right by 5 units.
### 2. Key Points on the Graph
By evaluating the function at specific [tex]\( x \)[/tex]-values, we can generate some key points that will help us sketch the graph.
Point 1:
- When [tex]\( x = 5 \)[/tex]:
[tex]\[
y = \sqrt[3]{5 - 5} = \sqrt[3]{0} = 0
\][/tex]
So, the point (5, 0) lies on the graph.
Point 2:
- When [tex]\( x = 6 \)[/tex]:
[tex]\[
y = \sqrt[3]{6 - 5} = \sqrt[3]{1} = 1
\][/tex]
So, the point (6, 1) lies on the graph.
Point 3:
- When [tex]\( x = 8 \)[/tex]:
[tex]\[
y = \sqrt[3]{8 - 5} = \sqrt[3]{3} \approx 1.442
\][/tex]
So, the point (8, 1.442) lies on the graph.
Point 4:
- When [tex]\( x = 12 \)[/tex]:
[tex]\[
y = \sqrt[3]{12 - 5} = \sqrt[3]{7} \approx 1.913
\][/tex]
So, the point (12, 1.913) lies on the graph.
### 3. Shape and Behavior of the Function
- The cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] is an odd function, showing symmetry around the origin, and this symmetry is retained but horizontally shifted to [tex]\( x = 5 \)[/tex].
- It increases monotonically since the cube root function always grows as [tex]\( x \)[/tex] increases.
- It passes through (5, 0) and continues to rise for [tex]\( x > 5 \)[/tex].
### 4. Sketching the Graph
Using the following key points:
- (5, 0)
- (6, 1)
- (8, 1.442)
- (12, 1.913)
We can sketch the graph:
1. Plot all the points on a coordinate plane.
2. Draw a smooth curve that starts at (5, 0) and asymptotically approaches the [tex]\( x \)[/tex]-axis as [tex]\( x \)[/tex] approaches 5 from the right.
3. Continue the curve upward through the plotted points, reflecting the increasing nature of the function.
### 5. Conclusion
To summarize, the graph of [tex]\( y = \sqrt[3]{x - 5} \)[/tex] is:
- Horizontally shifted right by 5 units compared to the basic cube root function.
- It passes through the calculated key points, rising as [tex]\( x \)[/tex] increases from the shift point at [tex]\( x = 5 \)[/tex].
- Given these characteristics, the correct graph will show an increasing curve starting from (5, 0), passing through (6, 1), (8, 1.442), and (12, 1.913).
