To determine which graph represents the function [tex]\( y = \sqrt[3]{x-5} \)[/tex], let’s break down the function and understand its characteristics and shape:
1. Understanding the Function:
- The function [tex]\( y = \sqrt[3]{x-5} \)[/tex] is a cube root function, with a horizontal shift 5 units to the right.
- The cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] is defined for all real numbers and has a certain shape that we need to shift horizontally by 5 units.
2. Key Points of the Function:
- The basic cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] passes through the origin [tex]\((0, 0)\)[/tex], but due to the shift, our function will pass through the point [tex]\((5, 0)\)[/tex].
- For [tex]\( x \)[/tex] values less than 5, [tex]\( y = \sqrt[3]{x-5} \)[/tex] will yield negative values as we are taking the cube root of a negative number.
- For [tex]\( x \)[/tex] values greater than 5, [tex]\( y = \sqrt[3]{x-5} \)[/tex] will yield positive values as we are taking the cube root of a positive number.
3. Shape Characteristics:
- The curve will cross the x-axis at [tex]\( x = 5 \)[/tex].
- For [tex]\( x > 5 \)[/tex], [tex]\( y \)[/tex] will be positive and increase, although not very steeply.
- For [tex]\( x < 5 \)[/tex], [tex]\( y \)[/tex] will be negative and decrease gradually.
4. Plot and Analysis:
- Start by plotting the key point [tex]\((5, 0)\)[/tex] where the function intersects the x-axis.
- Extend to the right side where the graph will gradually rise.
- Extend to the left side where the graph will gradually fall.
Based on these explanations, the correct graphical representation of the function [tex]\( y = \sqrt[3]{x-5} \)[/tex] will:
- Intersect the x-axis at the point [tex]\((5, 0)\)[/tex],
- Rise to the right of this point,
- Fall to the left of this point.
By comparing these characteristics with the given options, you should be able to identify the correct graph that fits these descriptions. Please examine the provided graphs or plots directly for the final identification.
