Question: Without graphing, determine whether the equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these.
y = x^2 + 6
Choose the correct answer below. Select all that apply.
A. x-axis
B. y-axis
C. origin
D. none of these
1. **State the problem:** Determine the symmetry of the graph of the equation $$y = x^2 + 6$$ without graphing.
2. **Recall symmetry rules:**
- Symmetry about the x-axis: Replace $$y$$ by $$-y$$ and check if the equation remains unchanged.
- Symmetry about the y-axis: Replace $$x$$ by $$-x$$ and check if the equation remains unchanged.
- Symmetry about the origin: Replace $$x$$ by $$-x$$ and $$y$$ by $$-y$$ and check if the equation remains unchanged.
3. **Test for x-axis symmetry:**
Replace $$y$$ by $$-y$$:
$$-y = x^2 + 6$$
Multiply both sides by $$-1$$:
$$y = -x^2 - 6$$
This is not the original equation, so no x-axis symmetry.
4. **Test for y-axis symmetry:**
Replace $$x$$ by $$-x$$:
$$y = (-x)^2 + 6 = x^2 + 6$$
This is the original equation, so the graph is symmetric about the y-axis.
5. **Test for origin symmetry:**
Replace $$x$$ by $$-x$$ and $$y$$ by $$-y$$:
$$-y = (-x)^2 + 6 = x^2 + 6$$
Multiply both sides by $$-1$$:
$$y = -x^2 - 6$$
This is not the original equation, so no origin symmetry.
**Final answer:** The graph is symmetric only about the y-axis.
**Answer choice:** B. y-axis