To identify the conic section represented by the equation \( y^2 - 3x + 4y + 7 = 0 \), we can rearrange it into a standard form.
Starting with the given equation:
\[ y^2 + 4y - 3x + 7 = 0 \]
We can isolate \( x \):
\[ 3x = y^2 + 4y + 7 \]
\[ x = \frac{1}{3}(y^2 + 4y + 7) \]
Next, let's analyze the form of the equation. In this equation, \( x \) is expressed as a function of \( y \), and it involves \( y^2 \) (which indicates that \( y \) is being squared).
Since this equation can be put into the form \( x = ay^2 + by + c \), where there is a squared term of \( y \) and the other terms are linear in \( y \), it is indicative of a parabola.
Thus, the conic section represented by the equation \( y^2 - 3x + 4y + 7 = 0 \) is:
Parabola
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