Many important curves cannot be represented globally as a single function \(y=f(x)\). A circle, for example, has two y-values for most x-values. Implicit equation mode lets you graph a relation written as
\[F(x,y)=0\]
without solving the entire relation for one variable.
1. Graph a circle
A circle centered at \((h,k)\) with radius \(r\) has equation
\[(x-h)^2+(y-k)^2=r^2.\]
For a circle centered at \((2,-1)\) with radius 3, use implicit mode and enter
(x-2)^2+(y+1)^2=9
A window such as \(-2\le x\le6\) and \(-5\le y\le3\) leaves enough margin around the curve.
2. Graph an ellipse
An axis-aligned ellipse has standard form
\[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.\]
Enter
(x-1)^2/4+(y+2)^2/9=1
to graph an ellipse centered at \((1,-2)\) with horizontal semiaxis 2 and vertical semiaxis 3. Its vertices are \((1,-5)\) and \((1,1)\); its co-vertices are \((-1,-2)\) and \((3,-2)\).
3. Why solving for y is often inconvenient
The circle equation \(x^2+y^2=9\) can be split into
\[y=\sqrt{9-x^2}\qquad\text{and}\qquad y=-\sqrt{9-x^2}.\]
This works, but it requires two function objects and a domain restriction \(-3\le x\le3\). The implicit equation is shorter and preserves the symmetry automatically.
4. Graph more general implicit curves
Implicit mode can also display curves such as
- \(x^3+y^3=3xy\), the folium of Descartes;
- \(x^2y+y^3=x\);
- \((x^2+y^2-1)^3-x^2y^3=0\), a heart-shaped algebraic curve.
Complex implicit curves may need a smaller grid step or higher quality setting to show narrow branches accurately.
5. Rotate a conic
A general quadratic relation may contain an \(xy\) term:
\[Ax^2+Bxy+Cy^2+Dx+Ey+F=0.\]
When \(B\ne0\), the principal axes are usually rotated relative to the coordinate axes. Implicit graphing avoids the lengthy algebra required to solve for y, although analytic classification still depends on the coefficients and discriminant \(B^2-4AC\).
6. Shade the inside of a conic
Replace equality by an inequality to describe a region. For example,
\[x^2+y^2\le9\]
is the closed disk of radius 3. Use the Graph Shading tool when you need color, opacity, or a region bounded by several curves.
Common mistakes
- Trying to enter a circle in ordinary \(y=f(x)\) mode.
- Forgetting parentheses around translated coordinates.
- Using unequal axis scales and misreading a circle as an ellipse.
- Choosing a viewing window that clips vertices or isolated components.
- Assuming every implicit curve passes the vertical-line test.
Open the Graphing Calculator and compare implicit, parametric, and explicit representations. For parameter-based conics, see parametric equations.