Find the equation of the parabola, which has vertex (0,0) and its symmetric about y-axis and passes through the point (2,-3).

Answer

To find the equation of the parabola with vertex at (0,0), symmetric about the y-axis, and passing through the point (2,-3), we can follow these steps:

### Step 1: Identify the standard form of the parabola
Since the parabola is symmetric about the y-axis and has its vertex at the origin (0,0), its equation can be written in the standard form:
\[ y = ax^2 \]
where \( a \) is a constant that determines the width and direction of the parabola.

### Step 2: Substitute the point into the equation
We know that the parabola passes through the point (2, -3). We will substitute \( x = 2 \) and \( y = -3 \) into the equation \( y = ax^2 \):
\[ -3 = a(2^2) \]
This simplifies to:
\[ -3 = 4a \]

### Step 3: Solve for \( a \)
Now, we will solve for \( a \):
\[ a = \frac{-3}{4} \]

### Step 4: Write the final equation of the parabola
Now that we have the value of \( a \), we can substitute it back into the standard form of the parabola:
\[ y = -\frac{3}{4}x^2 \]

### Final Answer
The equation of the parabola is:
\[ y = -\frac{3}{4}x^2 \]

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