The equation \( y^2 = 8x \) represents a parabola. Here are the key features of this parabola:
-
Orientation: The parabola opens to the right because the \( y^2 \) term is positive and it is expressed in the form \( y^2 = 4px \), where \( p \) is a positive constant.
-
Vertex: The vertex of the parabola is at the origin, \((0, 0)\).
-
Focus: The focus of the parabola can be determined using the formula \( p \) from the standard form \( y^2 = 4px \). In this case, \( 4p = 8 \), so \( p = 2 \). Therefore, the focus is located at \((2, 0)\).
-
Directrix: The directrix of the parabola is a vertical line located at \( x = -p \). Since \( p = 2\), the directrix is the line \( x = -2 \).
-
Axis of Symmetry: The axis of symmetry for this parabola is the x-axis (the line \( y = 0 \)), as the parabola is symmetric with respect to this line.
-
Latus Rectum: The length of the latus rectum, which is the line segment perpendicular to the axis of symmetry that passes through the focus, can be found using the formula \( 4p \). Here, it equals \( 8 \). The endpoints of the latus rectum are located at \( (2, 4) \) and \( (2, -4) \).
-
Intercepts:
- X-Intercept: The x-intercept occurs when \( y = 0 \), leading to \( 0 = 8x \). Therefore, the only x-intercept is at \((0, 0)\).
- Y-Intercept: The y-intercept occurs when \( x = 0 \), leading to \( y^2 = 0 \). So, the only y-intercept is also at \((0, 0)\).
Overall, the parabola \( y^2 = 8x \) is characterized by its rightward opening, vertex at the origin, focus at \((2, 0)\), and directrix line \( x = -2\).