On Functions and Their eraphs
Given an equation in and , how do you determine if its graph is symmetric with respect to the -axis?
A. A graph is symmetric with respect to the -axis if substituting - for in the equation results in an equivalent equation.
B. A graph is symmetric with respect to the -axis if substituting for and for in the equation results in an equivalent equation.
C. A graph is symmetric with respect to the -axis if substituting for in the equation results in an equivalent equation.

Ask by Clarke Salinas. in the United States

Jan 21,2025

Question

On Functions and Their eraphs
Given an equation in and , how do you determine if its graph is symmetric with respect to the -axis?
A. A graph is symmetric with respect to the -axis if substituting - for in the equation results in an equivalent equation.
B. A graph is symmetric with respect to the -axis if substituting for and for in the equation results in an equivalent equation.
C. A graph is symmetric with respect to the -axis if substituting for in the equation results in an equivalent equation.

Ask by Clarke Salinas. in the United States

Jan 21,2025

UpStudy AI · Accepted Answer

To determine if the graph of an equation in $$ x $$ and $$ y $$ is symmetric with respect to the $$ x $$-axis, you should:

**A. Substitute $$-y$$ for $$ y $$ in the equation and check if the resulting equation is equivalent to the original.**

**Explanation:**

- **Symmetry with respect to the $$ x $$-axis:**  
  Replace $$ y $$ with $$-y$$ in the equation. If the equation remains unchanged, then the graph is symmetric about the $$ x $$-axis.

- **Symmetry with respect to the $$ y $$-axis:**  
  Replace $$ x $$ with $$-x$$ in the equation. If the equation remains unchanged, then the graph is symmetric about the $$ y $$-axis.

- **Symmetry with respect to the origin:**  
  Replace both $$ x $$ with $$-x$$ and $$ y $$ with $$-y$$. If the equation remains unchanged, then the graph is symmetric about the origin.

**Example:**

Consider the equation $$ y = x^2 $$.

1. **Test for $$ x $$-axis symmetry:**  
   Replace $$ y $$ with $$-y$$:  
   $$ -y = x^2 $$  
   This is not equivalent to the original equation $$ y = x^2 $$.  
   Therefore, $$ y = x^2 $$ is **not** symmetric about the $$ x $$-axis.

2. **Test for $$ y $$-axis symmetry:**  
   Replace $$ x $$ with $$-x$$:  
   $$ y = (-x)^2 = x^2 $$  
   This is equivalent to the original equation.  
   Therefore, $$ y = x^2 $$ **is** symmetric about the $$ y $$-axis.

**Conclusion:**

Option **A** is the correct method to determine if a graph is symmetric with respect to the $$ x $$-axis.

**Answer:**  
A. A graph is symmetric with respect to the $$ x $$-axis if substituting –$$ y $$ for $$ y $$ in the equation results in an equivalent equation.
A graph is symmetric with respect to the $$ x $$-axis if replacing $$ y $$ with $$-y$$ in the equation results in an equivalent equation.

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