To identify which graph represents the equation [tex]\( y = \sqrt[3]{x} + 2 \)[/tex], let's break down the equation and analyze its characteristics:
1. General Shape: The term [tex]\( \sqrt[3]{x} \)[/tex] represents the cubic root function. The cubic root function [tex]\( \sqrt[3]{x} \modulates the function such that it produces a curve that is an odd function. Specifically, it will be an S-shaped curve because it's the inverse of the cube function \( y = x^3 \)[/tex]. The cubic root of a number can be both positive and negative; thus, it extends through all quadrants.
2. Vertical Shift: The entire function is shifted up by 2 units due to the [tex]\( +2 \)[/tex] term. This means the graph of [tex]\( y = \sqrt[3]{x} + 2 \)[/tex] will look like the graph of [tex]\( y = \sqrt[3]{x} \)[/tex] but moved up 2 units on the y-axis.
3. Behavior at Specific Points:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = \sqrt[3]{0} + 2 = 2 \)[/tex]. So, the graph will pass through the point [tex]\( (0, 2) \)[/tex].
- As [tex]\( x \)[/tex] increases or decreases significantly, the cubic root [tex]\( \sqrt[3]{x} \)[/tex] will also increase or decrease, but at a slower rate compared to linear or quadratic functions.
- For positive [tex]\( x \)[/tex], because [tex]\( \sqrt[3]{x} \)[/tex] is positive, the function [tex]\( y \)[/tex] will be [tex]\( 2 + \sqrt[3]{x} \)[/tex].
- For negative [tex]\( x \)[/tex], [tex]\( \sqrt[3]{x} \)[/tex] is negative, so [tex]\( y \)[/tex] will be less than 2 since we'll add a negative value to 2.
4. When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = \sqrt[3]{1} + 2 = 1 + 2 = 3 \quad \text{(point on the graph is \( (1, 3) \))}.
\][/tex]
5. When [tex]\( x = -1 \)[/tex]:
[tex]\[
y = \sqrt[3]{-1} + 2 = -1 + 2 = 1 \quad \text{(point on the graph is \( (-1, 1) \))}.
\][/tex]
6. Symmetry and Points of Interest: Since the cubic root function is odd, its positive and negative values are symmetric around the origin for [tex]\( \sqrt[3]{x} \)[/tex], and with the vertical shift applied, the new graph will be symmetric around the line [tex]\( y = 2 \)[/tex], though not exact point symmetry.
By assessing these key features, you can map out the graph's general behavior and verify it against the possible graph options presented to you. Given this analysis, you would look for a graph that:
- Passes through (0, 2)
- Is S-shaped, symmetric about [tex]\( y = 2 \)[/tex]
- Has particularly noted points like (1, 3), (-1, 1)
- Extends through all quadrants due to the nature of the cube root function.
Look for these attributes, and you will identify the correct graph.
