To determine which graph represents the function [tex]\( y = \sqrt[3]{x + 6} - 3 \)[/tex], let's break down the steps to understand the behavior of this function:
1. Understand the Basic Shape:
- The core part of the function is [tex]\( \sqrt[3]{x + 6} \)[/tex]. The cube root function, [tex]\( \sqrt[3]{x} \)[/tex], has an S-shaped curve that passes through the origin [tex]\((0,0)\)[/tex]. This function is defined for all real numbers and is symmetric with respect to the origin.
2. Horizontal Shift:
- The term [tex]\( x + 6 \)[/tex] means we are shifting the cube root function horizontally to the left by 6 units. Therefore, the point that would originally be at the origin [tex]\((0,0)\)[/tex] is now at the point [tex]\((-6, 0)\)[/tex].
3. Vertical Shift:
- The term [tex]\( -3 \)[/tex] outside the cube root shifts the entire graph downward by 3 units. So the point [tex]\((-6, 0)\)[/tex] is now moved to [tex]\((-6, -3)\)[/tex].
4. Plot Key Points:
- Let's find a few key points to help us sketch the graph:
- When [tex]\( x = -6 \)[/tex]:
[tex]\[
y = \sqrt[3]{-6 + 6} - 3 = \sqrt[3]{0} - 3 = 0 - 3 = -3
\][/tex]
- So we have the point [tex]\((-6, -3)\)[/tex].
- When [tex]\( x = -5 \)[/tex]:
[tex]\[
y = \sqrt[3]{-5 + 6} - 3 = \sqrt[3]{1} - 3 = 1 - 3 = -2
\][/tex]
- So we have the point [tex]\((-5, -2)\)[/tex].
- When [tex]\( x = -7 \)[/tex]:
[tex]\[
y = \sqrt[3]{-7 + 6} - 3 = \sqrt[3]{-1} - 3 = -1 - 3 = -4
\][/tex]
- So we have the point [tex]\((-7, -4)\)[/tex].
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
y = \sqrt[3]{2 + 6} - 3 = \sqrt[3]{8} - 3 = 2 - 3 = -1
\][/tex]
- So we have the point [tex]\( (2, -1) \)[/tex].
5. Behavior at Infinity:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( \sqrt[3]{x} \rightarrow \infty \)[/tex], so [tex]\( y = \sqrt[3]{x+6} - 3 \rightarrow \infty - 3 = \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( \sqrt[3]{x} \rightarrow -\infty \)[/tex], so [tex]\( y = \sqrt[3]{x+6} - 3 \rightarrow -\infty - 3 = -\infty \)[/tex].
6. Graph Characteristics:
- The graph will be continuous and will pass through our calculated points.
- It has an inflection point (a change in the curvature) at [tex]\((-6, -3)\)[/tex].
- For large negative [tex]\( x \)[/tex], the graph approaches negative infinity in a smooth, gradual curve.
- For large positive [tex]\( x \)[/tex], the graph gradually rises without bound.
Use these steps to identify which graph correctly displays these characteristics, ensuring that the origin point shift, each calculated data point, and the correct curvature are well-depicted.
