Introduction

Delving into the world of precalculus, implicitly defined function emerge as a fascinating topic to explore. Understanding these functions is crucial, especially in AP® Precalculus, as they lay the groundwork for more advanced mathematical concepts. Unlike explicitly defined functions, which are straightforward and predictable, implicitly defined functions hold a layer of complexity that challenges the usual method of finding relationships between variables. Therefore, grasping the basics of these functions and how they graphically represent data is essential for any high school precalculus student.

What is an Implicitly Defined Function?

An implicitly defined function is a function expressed through an equation involving two variables, without one being explicitly solved in terms of the other. In contrast, explicitly defined functions like y = f(x) provide a direct relationship where y is given in terms of x .

Example:

Consider the equation x^2 + y^2 = 25 . This equation characterizes a circle with a radius of 5. Unlike explicit functions, this equation does not isolate y in terms of x or x in terms of y . Instead, it defines a relationship between x and y that they jointly satisfy.

Graphing Implicitly Defined Functions

Graphing implicitly defined functions involves finding solutions to the equations by exploring various pairs of (x, y) that satisfy the given relationship. This process can identify symmetry and key features of the graph.

Example:

Graph the equation y^2 = x + 2y .

Step 1: Rearrange the Equation

Start by rearranging the equation if possible. For example, isolate terms: y^2 - 2y = x .

Step 2: Identify Key Points and Symmetry

Look for symmetry and easy points. Check x = 0 : y^2 = 2y \Rightarrow y(y-2) = 0 , so y = 0 or y = 2 . Key points are (0, 0) and (0, 2) .

Step 3: Sketch the Graph

Evaluate additional points, such as x =-1 , to see how y behaves. We can set x=-1 and solve for y again to find y=1 . Therefore, another point is (-1,1) Connect these with a smooth curve acknowledging any symmetry or specific features.

Understanding the Relationship Between Variables

Relationship Between Variables in an Implicit Function

Consider the equation of an ellipse: \frac{x^2}{9} + \frac{y^2}{4} = 1

This represents an ellipse centered at (0,0) with a horizontal axis of length 6 and a vertical axis of length 4.

To analyze how x and y change together, we look at nearby points:

• Point (0,2) → Moving slightly right to (1, ~1.89)
• Since increasing x makes y slightly decrease, the two variables change in opposite directions.
• Point (-3,0) → Moving slightly right to (-2, ~1.5)
• Here, y also increases, meaning the two variables change in the same direction.

This confirms that the ratio of change is negative when they move in opposite directions and positive when they move together.

Rate of Change

For the graph of x^2 + y^2 - xy = 4 on the x -interval (0,2) , what is the rate of change of y with respect to x ?

To find the rate of change of y with respect to x , we solve for two points on the curve.

Step 1: Find Two Points on the Graph

Let’s determine y for two values of x in the given interval.

1. When x = 0 : 0^2 + y^2 - (0)(y) = 4
   y^2 = 4
   Solutions: y = 2 or y = -2 .
2. When x = 2 : 2^2 + y^2 - (2)(y) = 4
   4 + y^2 - 2y = 4
   y^2 - 2y = 0
   y(y - 2) = 0 Solutions: y = 0 or y = 2 .

Step 3: Calculate the Rate of Change

Now, we have two points:

• At x = 0 , y = 2 or y = -2 .
• At x = 2 , y = 0 or y = 2 .

Step 4: Compute the Slope

Using the slope formula:

\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

For (0,2) and (2,0) :

\frac{0 - 2}{2 - 0} = \frac{-2}{2} = -1 .

For (0,-2) and (2,2) :

\frac{2 +2}{2 -0} = \frac{4}{2} = 2 .

Final Answer

The rate of change of y with respect to x depends on the branch of the curve.

Quick Reference Chart: Vocabulary and Definitions

Conclusion

Implicitly defined functions and their graphs present intriguing challenges in understanding real-world relationships between variables. Recognizing their importance in precalculus equips students with analytical tools essential for success in AP® Precalculus. With consistent practice and exploration, these concepts become less daunting and more intuitive. Students are encouraged to engage with various examples and problems to deepen understanding and proficiency.

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