Line Symmetry 940D11

1. **State the problem:** We need to determine the symmetry of the function given by the equation $$x + y = -4$$. 2. **Recall symmetry tests:** - **x-axis symmetry:** Replace $y$ by $-y$ and check if the equation remains the same. - **y-axis symmetry:** Replace $x$ by $-x$ and check if the equation remains the same. - **Origin symmetry:** Replace $x$ by $-x$ and $y$ by $-y$ and check if the equation remains the same. 3. **Test for x-axis symmetry:** Replace $y$ with $-y$: $$x + (-y) = -4 \implies x - y = -4$$ This is not the same as the original equation $x + y = -4$, so no x-axis symmetry. 4. **Test for y-axis symmetry:** Replace $x$ with $-x$: $$-x + y = -4$$ This is not the same as the original equation, so no y-axis symmetry. 5. **Test for origin symmetry:** Replace $x$ with $-x$ and $y$ with $-y$: $$-x + (-y) = -4 \implies -x - y = -4$$ Multiply both sides by $-1$ to compare: $$\cancel{-1}(-x - y) = \cancel{-1}(-4) \implies x + y = 4$$ This is not the same as the original equation $x + y = -4$, so no origin symmetry. 6. **Conclusion:** The function has none of the symmetries (x-axis, y-axis, or origin). **Final answer:** None of these.