4.5 Implicitly Defined Functions

An implicitly defined function comes from an equation in two variables like $x^2 + y^2 = 1$, where $x$ and $y$ are linked but neither is written alone. You graph it by finding all the $(x, y)$ pairs that make the equation true, and you can solve for one variable to pull out separate function pieces.

Why This Matters for the AP Precalculus Exam

Unit 4 topics, including implicitly defined functions, are not assessed on the AP Precalculus exam. The exam covers Units 1, 2, and 3. Still, this topic is worth learning because it strengthens skills you already use on the exam and sets you up for calculus.

Working with equations like circles, ellipses, and hyperbolas builds your fluency with multiple representations, which is tested heavily in earlier units. You practice moving between an equation, its graph, and a table of solutions. You also reason about how two quantities change together, which is a core idea behind covariation and rates of change that show up across the whole course. If your school teaches Unit 4, getting comfortable with these ideas now makes implicit differentiation in calculus much less intimidating later.

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Key Takeaways

• An equation in two variables can describe one or more functions at once; its graph is every (x, y) pair that makes the equation true.
• Solving for y (or for x) splits an implicit graph into separate function branches, like the top and bottom halves of a circle.
• Solving for a variable can also reveal the domain and range, and the domain may end up restricted compared to the original equation.
• For two nearby points on the curve, a positive change ratio means both variables move the same way; a negative ratio means one increases while the other decreases.
• When the rate of change of y with respect to x is zero, the curve is moving horizontally (horizontal interval).
• When the rate of change of x with respect to y is zero, the curve is moving vertically (vertical interval).

Graphing Equations Involving x and y

An equation involving two variables can implicitly describe one or more functions. The graph of the equation is the set of all solutions, meaning every coordinate pair (x, y) that makes the equation true.

For example, the equation $y = 2x + 1$ can be graphed by plotting all the solutions $(x, y)$ that make it true. You substitute different values of $x$ into the equation and solve for $y$.

The graph is a straight line that passes through (0, 1) with a slope of 2. This line is the graph of the function $y = 2x + 1$.

An equation in two variables can also describe a non-linear curve, such as a parabola or an ellipse. You graph these the same way: find solutions to the equation, then plot them on a coordinate plane.

Solving for One Variable

Solving for one of the variables can define a function whose graph is part or all of the graph of the equation.

Consider the equation $x^2 + y^2 = 1$. This describes the set of all points (x, y) that are the same distance from the origin, which is a circle with radius 1 centered at the origin.

If you solve for $y$, you get two functions, $y = \sqrt{1 - x^2}$ and $y = -\sqrt{1 - x^2}$. These describe the upper and lower halves of the circle. Graph them individually, and combined they form the full graph of $x^2 + y^2 = 1$.

Similarly, if you solve for $x$, you get $x = \sqrt{1 - y^2}$ and $x = -\sqrt{1 - y^2}$, which describe the right and left halves of the circle.

Solving for one of the variables also lets you find the domain and range of each function you get. In some cases, the domain of the function may be a restriction of the original equation, because some values of the variable you solved for will not satisfy the original equation.

Relating Functions Involving x and y

The graph of an implicitly defined function is the set of all ordered pairs (x, y) that satisfy the equation. For any two ordered pairs that are close together on the graph, the ratio of the change in the two variables (the slope of the line connecting the two points) tells you how the variables relate.

If the change ratio is positive, then as one variable increases the other increases too, and when one decreases the other decreases. If the ratio is negative, then as one variable increases the other decreases.

For example, take the circle $x^2 + y^2 = 1$ and two nearby points on it, such as (0.8, 0.6) and (0.9, 0.7). The change in $y$ over the change in $x$ is (0.7 - 0.6) / (0.9 - 0.8) = 1, so the $x$ and $y$ values increase together over that small step.

Horizontal and Vertical Intervals

The rate of change of one variable with respect to the other can be zero, and that signals special behavior on the curve.

When the rate of change of $y$ with respect to $x$ is zero, the curve is moving horizontally, and the slope between two nearby points is zero. This is a horizontal interval. ↔️

When the rate of change of $x$ with respect to $y$ is zero, the curve is moving vertically, and the slope between two nearby points is undefined. This is a vertical interval. ↕️

You can see this on $x^2 + y^2 = 1$. Take two nearby points (0.8, 0.6) and (0.78, 0.62) near the side of the circle. There the x-values barely change while the y-values move, which shows the curve heading vertically.

How to Use This on the AP Precalculus Exam

Unit 4 is not tested on the AP Precalculus exam, so there are no exam questions on implicitly defined functions. Use these strategies to build skills that transfer to tested units and to calculus.

Problem Solving

• When you see an equation in two variables, first decide whether to solve for $y$ or for $x$. Pick whichever isolates a variable cleanly.
• After solving, watch for the $\pm$ from a square root. Each sign is a separate function branch, so name both and know which part of the graph each one covers.
• Check the domain of every branch. A value that works for one branch may not satisfy the original equation.

Reading the Graph

• To tell how two quantities change together, pick two points close together and compute the change ratio. Positive means they move the same direction; negative means opposite directions.
• Look for points where the curve turns horizontal or vertical. A horizontal interval means the change in $y$ is zero over a small step; a vertical interval means the change in $x$ is zero.

Common Misconceptions

• An implicit equation is not automatically a single function. An equation like $x^2 + y^2 = 1$ fails the vertical line test, so it takes more than one function to describe its whole graph.
• A square root only gives the positive root by itself. To capture the full circle, you need both $y = \sqrt{1 - x^2}$ and $y = -\sqrt{1 - x^2}$.
• Horizontal and vertical intervals get swapped easily. A horizontal interval happens when the change in $y$ is zero (slope 0); a vertical interval happens when the change in $x$ is zero (slope undefined).
• The change ratio between two points is a slope, not a guarantee about the whole curve. It describes the relationship only over that small step, and it can change sign at different places on the graph.
• The domain of a branch is not always the same as the original equation. Solving for a variable can restrict which inputs actually produce valid points.

Related AP Precalculus Guides

• 4.10 Matrices
• 4.12 Linear Transformations and Matrices
• 4.6 Conic Sections
• 4.9 Vector-Valued Functions
• 4.13 Matrices as Functions
• 4.4 Parametrically Defined Circles and Lines