Sure! Let's solve the question involving the given function [tex]\( y = x^3 + 2 \)[/tex]. We'll break this down into the components involved in the expression:
1. Identification of the function:
- We have a function [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], which is given by [tex]\( y = x^3 + 2 \)[/tex].
2. Analyzing the Components:
- [tex]\( x^3 \)[/tex] is a cubic term. This suggests that the function is a cubic function, which generally has a curve that starts from one end and diverges significantly at the other end.
- The [tex]\( + 2 \)[/tex] is a constant that shifts the graph of the cubic function [tex]\( x^3 \)[/tex] vertically upwards by 2 units.
3. Behavior of the Function:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
4. Key Points:
- To plot or understand this function graphically, let's find some key points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0^3 + 2 = 2
\][/tex]
So, we have the point (0, 2).
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 1^3 + 2 = 3
\][/tex]
So, we have the point (1, 3).
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
y = (-1)^3 + 2 = -1 + 2 = 1
\][/tex]
So, we have the point (-1, 1).
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 2^3 + 2 = 8 + 2 = 10
\][/tex]
So, we have the point (2, 10).
- When [tex]\( x = -2 \)[/tex]:
[tex]\[
y = (-2)^3 + 2 = -8 + 2 = -6
\][/tex]
So, we have the point (-2, -6).
5. Graphical Representation:
- If you plot these points on a graph, you will see that the function [tex]\( y = x^3 + 2 \)[/tex] forms a curve typical of cubic functions that crosses the y-axis at (0, 2) and moves upwards as [tex]\( x \)[/tex] increases, and downwards as [tex]\( x \)[/tex] decreases.
6. Conclusion:
- The function [tex]\( y = x^3 + 2 \)[/tex] describes a cubic curve shifted up by 2 units. It passes through crucial points like (0, 2), (1, 3), (-1, 1), (2, 10), and (-2, -6).
To summarize, the function [tex]\( y = x^3 + 2 \)[/tex] behaves as a cubic curve shifted upwards by 2 units and will have a similar cubic shape, but starting higher on the y-axis.
