91Ó°ÊÓ

In algebra, a parabola's equation can be expressed in standard forms that show whether it opens upwards, downwards, to the left, or to the right. For parabolas opening horizontally, the standard form of the equation is\[ y^2 = 4px \]where:- \( y^2 \) shows the parabola is opening either left or right depending on the sign of \( p \).- \( p \) represents the distance from the vertex to the focus.- The parabola's vertex is at the origin
For the equation \( y^2 = -4x \), it is in standard form and indicates a parabola that opens to the left. The negative sign before \( 4x \) suggests that the parabola will open in the negative x-direction.

The vertex and the focus are crucial in understanding a parabola's graph. The vertex is the 'turning point' of the parabola, where it changes direction. For the equation \(y^2 = -4x\), the vertex is at point The focus of a parabola is a point inside the curve where all the points are equidistant from the directrix. It helps in describing the shape and stretch of the parabola.- With a vertex at Using the standard form equation \(y^2 = 4px\), and knowing that \(4p = -4\), we find that \(p = -1\). Thus, the focus is located at \((-1, 0)\), one unit to the left of the origin.- The directrix, on the other hand, is a line equidistant from the vertex as the focus but in the opposite direction, described by \(x = 1\).

One of the defining properties of a parabola is its symmetry. Parabolas have an axis of symmetry, which is a line that divides the parabola into two mirror-image halves. For the horizontal parabola given by \(y^2 = -4x\), the axis of symmetry is the x-axis itself.Understanding symmetry helps in graphing the parabola, as the curve will be a mirror image on both sides of this axis.- The left opening of the parabola, in this case, reinforces that each y-value (positive or negative) has a corresponding opposite y-value,Knowing a parabola's axis of symmetry brings simplicity to plotting multiple key points of the graph, as these points will be repeated on either side, ensuring an accurate and balanced representation of the curve.