Reference Tables for Pre-Calculus
Conic Sections Circles
The standard form of the equation of a circle is:
(x - h)2 + (y - k)2 = r2
where:
(h, k) is the center of the circle r is the radius
The general form circle is: x2 + y2 + Dx + Ey + F = 0
Note that the leading coefficients of x2 and y2 are equal and there are no terms that contain xy
Ellipses
The standard form of the equation of an ellipse centered at (h, k) and with foci parallel to or lying on the x-axis is:
where:
foci = (h ± c, k) c2 = a2 – b2
The standard form of the equation of an ellipse centered at (h, k) and with foci parallel to or lying on the y-axis is:
where:
foci = (h, k ± c) c2 = a2 – b2
The general form of the equation of an ellipse with axes parallel to the coordinate axes is: Ax2 + Cy2 + Dx + Ey + F = 0
The endpoints of the major axes are:
(h ± a, k) if the major axis is horizontal or
(h, k ± a) if the major axis is vertical
The endpoints of the minor axes are:
(h ± b, k) if the minor axis is horizontal or
(h, k ± b) if the minor axis is vertical
Eccentricity of an ellipse:
Hyperbolas
The standard form of the equation of a hyperbola with a horizontal transverse axis is:
where:
(h, k) is the center of the hyperbola
The vertices are (h ± a, k)
The foci are (h ± c, k) where c2 = a2 + b2
The length of the transverse axis is 2a
The length of the conjugate axis is 2b
The endpoints of the conjugate axis are (h, k ± b)
The asymptotes are
The standard form of the equation of a hyperbola with a vertical transverse axis is:
where:
(h, k) is the center of the hyperbola
The vertices are (h, k ± a)
The foci are (h, k ± c) where c2 = a2 + b2
The length of the transverse axis is 2a
The length of the conjugate axis is 2b
The endpoints of the conjugate axis are (h ± b, k)
The asymptotes are
Parabolas
The equation of a parabola with a vertex of (0, 0), a focus of (0, c), a directrix of y = -c and the y-axis as the axis of symmetry is:
The equation of a parabola with a vertex of (0, 0), a focus of (c, 0), a directrix of x = -c and the x-axis as the axis of symmetry is:
The standard form of the equation of a parabola centered at (h, k) with a vertical axis of symmetry is:
(x - h)2 = 4c(y - k)
where:
The vertex is (h, k)
The axis of symmetry is x = h
The focus is (h, k + c)
The directrix is y = k - c
Opening:
Upward if c > 0
Downward if c < 0
The general form is: Ax2 + Dx + Ey + F = 0
The standard form of the equation of a parabola centered at (h, k) with a horizontal axis of symmetry is:
(y - k)2 = 4c(x - h)
where:
The vertex is (h, k)
The axis of symmetry is y = k
The focus is (h + c, k)
The directrix is x = h - c
Opening
Right if c > 0
Left if c < 0
The general form is: Cy2 + Dx + Ey + F = 0
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