AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/4/4-5)
Unit 4 · Topic 4.5
4.5 Implicitly Defined Functions
An equation in x and y, like x² + y² = 25, can describe a curve without giving y as a single function of x. Such a curve is implicitly defined. You'll graph these curves, split them into functions by solving for one variable, and describe how x and y change together.
Key terms
- implicitly defined function
- explicit function
- relation
- rate of change
Implicit vs explicit
An explicit function gives one variable directly in terms of the other, like y = 3x − 1. An implicitly defined curve is described by an equation relating x and y, like x² + y² = 25 or xy = 6, without solving for either.
An implicit equation can describe one function, several functions, or a curve that is not a function of x at all. The circle x² + y² = 25 fails the vertical line test, but it is made of two functions.
Other familiar examples: xy = 6 describes two separate pieces, and each one is part of the function y = 6/x. The ellipses and hyperbolas in topic 4.6 are also implicitly defined.
To test whether a point is on an implicit curve, substitute its coordinates. (3, −4) is on x² + y² = 25 because 9 + 16 = 25, and (3, 3) is not because 9 + 9 = 18.
Graphing by finding solutions
The graph is every point (x, y) that makes the equation true. To find points, pick a value for one variable and solve for the other.
For x² + y² = 25: x = 0 gives y = ±5, x = 3 gives y = ±4, and x = 5 gives y = 0. These points sketch a circle of radius 5.
Solving for one variable
Solving for y (or x) can split the curve into functions. From x² + y² = 25, y = √(25 − x²) is the top half and y = −√(25 − x²) is the bottom half. Each is a function on −5 ≤ x ≤ 5.
Sometimes it's easier to solve for x as a function of y. For x = y² − 4y, the curve is a function of y but not of x.
How x and y change together
For two nearby points on the curve, look at the ratio (change in y)/(change in x). If it's positive, x and y increase together or decrease together. If it's negative, one increases while the other decreases.
If the change in y is zero for nearby points, the curve is flat there: a horizontal piece, like the top of a circle. If the change in x is zero, so (change in x)/(change in y) is zero, the curve is vertical there, like the left and right edges of a circle.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Splitting a circle into functions
Write x² + y² = 25 as two functions of x, give their domains, and find the points where x = 4.
Show the solutionHide the solution
- Step 1: Solve for y: y² = 25 − x², so y = √(25 − x²) or y = −√(25 − x²).
- Step 2: The expression under the root must be at least 0: 25 − x² ≥ 0, so −5 ≤ x ≤ 5 for both.
- Step 3: At x = 4: y = ±√9 = ±3, so the points are (4, 3) on the top function and (4, −3) on the bottom.
Answer: y = √(25 − x²) and y = −√(25 − x²), each on −5 ≤ x ≤ 5; the points (4, 3) and (4, −3).
- Example 2
A sideways parabola
Graph x = y² − 4y by finding points, and write y in terms of x.
Show the solutionHide the solution
- Step 1: Pick y-values: y = 0 gives x = 0; y = 1 gives x = −3; y = 2 gives x = −4; y = 3 gives x = −3; y = 4 gives x = 0.
- Step 2: The points form a parabola opening to the right with vertex (−4, 2). The curve is vertical at the vertex.
- Step 3: Complete the square: x = (y − 2)² − 4, so (y − 2)² = x + 4 and y = 2 ± √(x + 4), for x ≥ −4.
Answer: A right-opening parabola with vertex (−4, 2); y = 2 + √(x + 4) (top half) and y = 2 − √(x + 4) (bottom half).
- Example 3
Trap: the sign of the change depends on where you are
On the circle x² + y² = 25, compare the change ratios from (3, 4) to (4, 3) and from (−4, 3) to (−3, 4).
Show the solutionHide the solution
- Step 1: From (3, 4) to (4, 3): change in y is −1 and change in x is 1, ratio −1. As x increases, y decreases.
- Step 2: From (−4, 3) to (−3, 4): change in y is 1 and change in x is 1, ratio 1. Both increase together.
- Step 3: The same equation gives different behavior on different parts of the curve. Don't describe the whole curve with one sign.
Answer: The ratio is −1 near the upper right and 1 near the upper left of the circle.
Common mistakes
- Forgetting the ± when taking a square root to solve for y, which loses half the curve.
- Assuming an implicit curve must be a function of x.
- Describing how x and y change together for the whole curve at once, when it varies from piece to piece.
On the exam
- Unit 4 is not on the AP Precalculus Exam, so you'll see this topic on class tests rather than in May.
- Expect to find points on an implicit curve, solve for one variable to get functions, and describe where the curve is horizontal or vertical. Implicit equations return in AP Calculus.
Connected topics
Videos
Check yourself
3 questions on 4.5 Implicitly Defined Functions. Pick an answer to see if you got it, and why.
The graph of x² + y² = 25 is a circle. Which function's graph is the lower half of the circle?
The graph of x² + 4y² = 16 is an ellipse. Which function's graph is the upper half of the ellipse?
The graph of x² + y² − 6y = 0 is not the graph of a function of x. Which pair of functions together have the same graph?
0 of 3 answered