Decide whether the graph of each equation is symmetric with respect to the origin. y=x^(2)+4x+2

To determine if the graph of the equation \( y = x^2 + 4x + 2 \) is symmetric with respect to the origin, we need to test its symmetry properties.

### Symmetry with Respect to the Origin
A graph is symmetric with respect to the origin if replacing \( x \) with \(-x\) and \( y \) with \(-y\) results in the same equation.

1. **Original Equation:**  
   \[
   y = x^2 + 4x + 2
   \]

2. **Replace \( x \) with \(-x\) and \( y \) with \(-y\):**  
   Substitute \(-x\) into the equation:  
   \[
   y = (-x)^2 + 4(-x) + 2
   \]
   Simplifying this:  
   \[
   y = x^2 - 4x + 2
   \]

3. **Replace \( y \) with \(-y\):**  
   Now substitute \(-y\):  
   \[
   -y = x^2 - 4x + 2
   \]
   Or:  
   \[
   y = -(x^2 - 4x + 2)
   \]
   Simplifying further:  
   \[
   y = -x^2 + 4x - 2
   \]

Since the transformed equation \( y = -x^2 + 4x - 2 \) does not match the original equation \( y = x^2 + 4x + 2 \), the graph is **not symmetric with respect to the origin**.

### Summary
- **Symmetry with respect to the y-axis?** No
- **Symmetry with respect to the x-axis?** No
- **Symmetry with respect to the origin?** No

The given function \( y = x^2 + 4x + 2 \) does not have symmetry with respect to the origin.

Would you like further clarification, or do you have any questions?

### Related Questions:
1. How do you test for symmetry with respect to the y-axis?
2. How do you find the vertex of a parabola given its equation?
3. How does the symmetry of a graph relate to its algebraic properties?
4. What are the general rules for testing symmetry in functions?
5. How would you graph the equation \( y = x^2 + 4x + 2 \)?

**Tip:** Symmetry tests are great for quickly checking if certain transformations of variables will change the function.

Decide whether the graph of each equation is symmetric with respect to the origin. y = x^2 + 4x + 2

Determine whether the quadratic equation y = x^2 + 4x + 2 is symmetric with respect to the origin. This step-by-step guide explains how to test for symmetry by replacing variables, covering key algebraic concepts for high school students in Grades 9-12.

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