Certainly! Let's analyze the function [tex]\( y = \sqrt[3]{x+6} - 3 \)[/tex] step by step to help identify which graph represents this equation.
1. Function Form: The given function is [tex]\( y = \sqrt[3]{x+6} - 3 \)[/tex]. This is a transformation of the basic cube root function [tex]\( y = \sqrt[3]{x} \)[/tex].
2. Horizontal Shift: The term [tex]\( x+6 \)[/tex] inside the cube root denotes a horizontal shift. Specifically, the graph of [tex]\( y = \sqrt[3]{x} \)[/tex] is shifted 6 units to the left.
3. Vertical Shift: After the cube root, subtracting 3 from the result represents a vertical shift. Thus, the graph is shifted 3 units downward.
4. Behavior of Cube Root Function: The basic cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] has the following key characteristics:
- It is defined for all real values of [tex]\( x \)[/tex].
- It passes through the origin [tex]\((0,0)\)[/tex] in its unshifted form.
- For positive [tex]\( x \)[/tex], [tex]\( y \)[/tex] gradually increases.
- For negative [tex]\( x \)[/tex], [tex]\( y \)[/tex] gradually decreases.
- It has a point of inflection at [tex]\( x = 0 \)[/tex].
5. Critical Point Analysis: To identify specific critical points moved by the transformations:
- Compute the shifted origin. For the cube root function, originally passing through (0,0), shifted left by 6 and down by 3, the new point becomes:
[tex]\( (0-6, 0-3) = (-6, -3) \)[/tex].
- Identify any intercept with the y-axis: Substituting [tex]\( x = 0 \)[/tex] in the function:
[tex]\[
y = \sqrt[3]{0+6} - 3 = \sqrt[3]{6} - 3.
\][/tex]
Thus, the y-intercept is at [tex]\( (0, \sqrt[3]{6} - 3) \)[/tex].
By summarizing these shifts and transformations, visual characteristics can guide us to identify the correct graph for [tex]\( y = \sqrt[3]{x+6} - 3 \)[/tex]:
- A leftward (negative x) shift by 6 units
- A downward (negative y) shift by 3 units
- A smooth curve that continues in both positive and negative directions of x
Remember, the result showed that x = 723 is a solution for setting [tex]\( y = 0 \)[/tex]. Checking this helps confirm a qualitative property if graph matches this intercept positioning.
By these clues and interpretations, choose a graph which portrays a cubic root shape with indicated transformations and features.
