Graph PlottersExplore secant branches and vertical asymptotes while changing scale, frequency, phase, and vertical shift.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
Secant is the reciprocal of cosine, with repeating branches separated by vertical asymptotes.
HOW TO USE IT
Secant is the reciprocal of cosine: sec(x) = 1/cos(x). Its graph has repeating U-shaped branches and vertical asymptotes where cosine equals zero. In y = A sec(Bx + C) + D, the coefficients control scale, period, horizontal position, and vertical shift.
The secant function has period 2π/|B| in radians or 360°/|B| in degrees. Unlike sine and cosine, secant is not limited between −1 and 1. Its branches lie above or below the midline according to the scale and shift, with gaps at the asymptotes.
Use the number fields or quick examples to change A, B, C, and D. Compare the branch spacing and asymptotes after each change, then use the graph cursor to inspect points away from undefined locations.
Secant is the reciprocal of cosine, so sec(x) = 1/cos(x). It is undefined at cosine zeros and its graph has U-shaped branches with ranges y ≤ −1 or y ≥ 1 before transformations. In y = A sec(Bx + C) + D, the reciprocal structure remains visible while A changes the branch distance from the midline, B changes repetition, C moves the asymptotes, and D shifts the center. A secant graph is easiest to understand beside its cosine parent.
For y = 2 sec(x) − 1, the period is 2π, the midline is y = −1, and the branches lie at or beyond y = 1 and y = −3. Plot the equation, compare its gaps with the zeros of cosine, and avoid treating the asymptotes as ordinary points. Use the graph cursor on a branch to confirm that reciprocal values grow as cosine approaches zero.
For y = sec(x) + 2, the period remains 2π and the midline is y = 2. The branch ranges become y ≤ 1 or y ≥ 3, while the vertical asymptotes stay at the zeros of cosine. Plot the equation next to the cosine idea in your notes: where cosine is close to zero, secant becomes large in magnitude, creating the characteristic gaps and branches.
Because sec(x) is the reciprocal of cosine and cosine lies between −1 and 1, secant satisfies sec(x) ≤ −1 or sec(x) ≥ 1. After a transformation y = A sec(Bx + C) + D, the branch boundaries become D − |A| and D + |A|. These values provide a quick check for whether a plotted branch is in the correct vertical position.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
Secant is the reciprocal of cosine: sec(x) = 1/cos(x).
The period is 2π/|B| in radians or 360°/|B| in degrees.
Secant is undefined wherever cosine is zero.
Secant is undefined wherever cos(x) = 0, because sec(x) = 1/cos(x). Those locations become vertical asymptotes.
No. Secant is unbounded. The coefficient A changes branch scale and the nearest branch values rather than setting a maximum height.
The period is 2π/|B| in radians or 360°/|B| in degrees, matching the cosine input frequency.