AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.5 — Implicitly Defined Functions
Notes — When y Isn't a Function of x
💡 Learning Objectives (4.5.A)
By the end of this lesson you will be able to:
- Distinguish between explicit and implicit forms of an equation in x and y
- Identify whether a given equation defines y as a function of x
- Sketch graphs of implicit equations by point-plotting and symmetry
- Find intercepts and key features of an implicit equation
1. Explicit vs. Implicit
An EXPLICIT equation gives y directly as a function of x: y = some expression in x. Examples: y = x², y = sin x, y = ln(x + 1).
An IMPLICIT equation relates x and y in a more general way, without isolating y. Examples: x² + y² = 25, x² + xy + y² = 7, sin(xy) = 1/2.
Many implicit equations describe curves that are NOT functions — they fail the vertical line test (more than one y-value for a given x).
2. Common Implicit Curves
- Circle: x² + y² = r² (centered at origin)
- Ellipse: x²/a² + y²/b² = 1
- Hyperbola: x²/a² − y²/b² = 1
- Parabola (sideways): y² = 4px
- More exotic: x³ + y³ = 6xy (folium of Descartes)
3. Determining if y Is a Function of x
Apply the VERTICAL LINE TEST: if ANY vertical line crosses the curve at more than one point, y is NOT a function of x.
📘 Example — Function or not?
- y = x² + 3: yes, function (it's already explicit and one y per x)
- x² + y² = 25: NOT a function — for x = 0, y can be ±5
- y = ±√(25 − x²): an implicit relation that we can split into two functions, y = √(25 − x²) (top half) and y = −√(25 − x²) (bottom half)
4. Sketching Implicit Equations
Even when the equation isn't explicitly y =, you can still sketch the curve. Strategies:
- Plug in convenient x or y values to find points
- Find intercepts: set y = 0 to find x-intercepts; set x = 0 for y-intercepts
- Check for symmetry: replace (x, y) with (−x, y), (x, −y), or (−x, −y) and see if the equation is unchanged
- Look for asymptotic behavior at large |x| or |y|
- Identify the curve type if it matches a standard form (circle, ellipse, etc.)
5. Symmetry Tests
💡 Three Symmetry Tests for Implicit Curves
- Symmetric about y-axis: replace x with −x; if equation unchanged, yes
- Symmetric about x-axis: replace y with −y; if unchanged, yes
- Symmetric about origin: replace (x, y) with (−x, −y); if unchanged, yes
📘 Example — Symmetries of x² + y² = 25
- Replace x with −x: (−x)² + y² = 25 ⇒ same equation ✓ (y-axis symmetry)
- Replace y with −y: x² + (−y)² = 25 ⇒ same ✓ (x-axis symmetry)
- Both ⇒ origin symmetry too
- Confirms what we know: a circle has all three symmetries.
6. Solving for y to Make Functions
Sometimes you can split an implicit equation into multiple explicit functions:
📘 Example — Split x² + y² = 25
- Solve for y: y² = 25 − x², so y = √(25 − x²) or y = −√(25 − x²)
- Top half: y = √(25 − x²) — a function with domain [−5, 5]
- Bottom half: y = −√(25 − x²) — also a function
- Together they form the full circle
7. Finding Specific Points
Plug in a specific x and solve for y (or vice versa):
📘 Example — Points on x² + xy + y² = 7
- Set x = 1: 1 + y + y² = 7 ⇒ y² + y − 6 = 0 ⇒ (y + 3)(y − 2) = 0
- So y = −3 or y = 2; the points (1, 2) and (1, −3) are on the curve
- Notice — TWO different y-values for x = 1, so the curve is not a function.
8. Summary
- Implicit equations relate x and y without solving for y
- Use the vertical line test to determine whether the curve is a function
- Find points by plugging in specific x or y; find intercepts by setting one to 0
- Three symmetry tests speed up sketching
- An implicit equation can sometimes be split into multiple explicit functions