find the equation of parabola which is symmetric about y-axis,and passes through point (2,-3).

Explanation

To find the equation of a parabola that is symmetric about the y-axis and passes through the point (2, -3), we can follow these steps:

### Step 1: Understand the general form of the parabola
Since the parabola is symmetric about the y-axis, its equation can be expressed in the form:
\[ x^2 = 4ay \]
or
\[ x^2 = -4ay \]
For our case, since the parabola opens downwards, we will use:
\[ x^2 = -4ay \]

### 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:
\[ 2^2 = -4a(-3) \]

### Step 3: Simplify the equation
Calculating the left side:
\[ 4 = -4a(-3) \]
This simplifies to:
\[ 4 = 12a \]

### Step 4: Solve for \(a\)
Now, we can solve for \(a\):
\[ a = \frac{4}{12} = \frac{1}{3} \]

### Step 5: Write the equation of the parabola
Now that we have the value of \(a\), we can substitute it back into the equation of the parabola:
\[ x^2 = -4 \left(\frac{1}{3}\right) y \]
This simplifies to:
\[ x^2 = -\frac{4}{3}y \]

### Step 6: Rearranging the equation
To express the equation in a standard form, we can rearrange it:
\[ x^2 + \frac{4}{3}y = 0 \]

### Final Answer
Thus, the equation of the parabola is:
\[ x^2 + \frac{4}{3}y = 0 \]

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