Create a similar question which Symmetry with Respect to the X-axis and find its equation

Sure! Let's create a similar question where the parabola is symmetric with respect to the x-axis and find its equation.

**Question:**
Find the equation of the parabola with vertex at the origin, passing through the point \( P(3, 4) \) and symmetric with respect to the x-axis.

**Solution:**

For a parabola symmetric with respect to the x-axis, the general form is:

\[ x = ay^2 \]

Given that the parabola passes through \( P(3, 4) \), we substitute \( x = 3 \) and \( y = 4 \) into the equation to find the value of \( a \):

\[ 3 = a(4^2) \]
\[ 3 = 16a \]
\[ a = \frac{3}{16} \]

Thus, the equation of the parabola is:

\[ x = \frac{3}{16}y^2 \]

To write this equation in a form similar to multiple-choice options, we multiply both sides by 16:

\[ 16x = 3y^2 \]
\[ 3y^2 - 16x = 0 \]

So, the equation of the parabola is:

\[ \boxed{3y^2 - 16x = 0} \]

Would you like more details or have any questions about this solution?

Here are 5 questions you might want to ask:
1. What is the general form of a parabola symmetric about the x-axis?
2. How can you derive the equation of a parabola given a point and its vertex?
3. What are the properties of parabolas symmetric with respect to the x-axis?
4. How do we handle parabolas that open downward or to the left?
5. What are the differences between parabolas symmetric with respect to the y-axis and x-axis?

**Tip:** When dealing with parabolas, always pay attention to their symmetry and orientation as it dictates the general form of the equation.

Learn how to find the equation of a parabola that is symmetric with respect to the x-axis, using a detailed step-by-step solution. This problem covers key concepts in parabolas, symmetry, and quadratic equations. Suitable for high school students.

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