Graph PlottersPlot 1/x and transformed reciprocal functions with live asymptotes, domain, and intercept information.
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
The parent reciprocal function y = 1/x has two branches separated by vertical and horizontal asymptotes.
HOW TO USE IT
The parent reciprocal function y = 1/x has two branches and is undefined at x = 0. This calculator uses y = A/(Bx + C) + D so you can explore scale, asymptotes, horizontal movement, and vertical movement in one workspace.
When B is nonzero, the vertical asymptote is x = −C/B and the horizontal asymptote is y = D. The graph never crosses its asymptotes in the parent form. A negative A changes which quadrants contain the branches.
Use reciprocal examples to see how the numerator scale changes distance from the asymptotes. Hover over a branch, use zoom near an intercept, and remember that the function is not defined at its vertical asymptote.
The parent reciprocal function y = 1/x is undefined at x = 0 and has two branches approaching the vertical and horizontal asymptotes. In y = A/(Bx + C) + D, the vertical asymptote is x = −C/B and the horizontal asymptote is y = D when B is nonzero. A controls how far the branches sit from their asymptotes, while its sign determines which pair of regions contains them.
For y = 2/(x − 1) + 3, the asymptotes are x = 1 and y = 3. The graph is undefined at x = 1 and approaches 3 as x becomes very large in either direction. Plot the example, zoom near the center, and test x = 0 to get y = 1. Checking the asymptotes first prevents a common mistake: treating the missing point as a removable hole or an ordinary intercept.
For y = 3/(x + 2) − 1, the vertical asymptote is x = −2 and the horizontal asymptote is y = −1. At x = 1, the value is 0, so the graph has an x-intercept at (1, 0). Enter those values and use the cursor to check the intercept while remembering that the curve cannot be evaluated at x = −2.
A reciprocal graph has a vertical asymptote when the denominator is zero and the function grows without bound nearby. A removable hole is different: it occurs when a common factor cancels and leaves a missing point with a finite limiting value. For the calculator’s basic reciprocal form, focus on the asymptotes and branch behavior rather than treating the gap as an ordinary point.
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
The parent reciprocal function is y = 1/x.
For A/(Bx + C) + D, it is x = −C/B when B is nonzero.
The horizontal asymptote is y = D.
For the basic form A/(Bx + C) + D, the graph does not equal D because the reciprocal term cannot be zero.
Set the denominator equal to zero. For A/(Bx + C) + D, solve Bx + C = 0.
The denominator changes sign on either side of the vertical asymptote, placing the reciprocal values in two separate regions.