Respuesta :
Answer:
- 4. no origin symmetry
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Given function:
- y = x³ + 5
Graph it first (see attached).
Test the graph for symmetry.
We know the odd degree parent function y = x³ has an origin symmetry, whilst even degree functions may have axis symmetry.
The given function is a translation of cubic function 5 units up so the center of symmetry has translated as well. Therefore correct answer is 4.

Answer:
4. no origin symmetry
Step-by-step explanation:
Functions are symmetric with respect to the x-axis if for every point (a, b) on the graph, there is also a point (a, −b) on the graph:
- f(x, y) = f(x, −y)
To determine if a graph is symmetric with respect to the x-axis, replace all the y's with (−y). If the resultant expression is equivalent to the original expression, the graph is symmetric with respect to the x-axis.
[tex]\begin{aligned}&\textsf{Given}: \quad &y &= x^3 + 5\\&\textsf{Replace $y$ for $(-y)$}: \quad &-y &= x^3 + 5\\&\textsf{Simplify}: \quad &y &= -x^3 - 5\end{aligned}[/tex]
Therefore, since the resultant expression is not equivalent to the original expression, it is not symmetric with respect to the x-axis.
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Functions are symmetric with respect to the y-axis if for every point (a, b) on the graph, there is also a point (-a, b) on the graph:
- f(x, y) = f(-x, y)
To determine if a graph is symmetric with respect to the x-axis, replace all the x's with (−x). If the resultant expression is equivalent to the original expression, the graph is symmetric with respect to the y-axis.
[tex]\begin{aligned}&\textsf{Given}: \quad &y&=x^3+5\\&\textsf{Replace $x$ for $(-x)$}: \quad &y&=(-x)^3+5\\&\textsf{Simplify}: \quad &y&=-x^3+5\end{aligned}[/tex]
Therefore, since the resultant expression is not equivalent to the original expression, it is not symmetric with respect to the y-axis.
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Functions are symmetric with respect to the origin if for every point (a, b) on the graph, there is also a point (-a, -b) on the graph:
- f(x, y) = f(-x, -y)
To determine if a graph is symmetric with respect to the origin, replace all the x's with (−x) and all the y's with (-y). If the resultant expression is equivalent to the original expression, the graph is symmetric with respect to the origin.
[tex]\begin{aligned}&\textsf{Given}: \quad &y&=x^3+5\\&\textsf{Replace $x$ for $(-x)$ and $y$ for $(-y)$}: \quad &(-y)&=(-x)^3+5\\&\textsf{Simplify}: \quad &-y&=-x^3+5\\&&y&=x^3-5\end{aligned}[/tex]
Therefore, since the resultant expression is not equivalent to the original expression, it is not symmetric with respect to the origin.
