4.5 Implicitly defined functions
Introduction
Learning objective
- 4.5.A - Construct a graph of an equation involving two variables.
- 4.5.B - Determine how the two quantities related in an implicitly defined function vary together.
Graphs of implicity defined functions
An implicitly defined function is an equation involving two variables that can describe one or more functions without explicitly stating the dependent variable (output) in terms of the independent variable (input). Implicitly defined functions can be a powerful way to represent relationships between variables, as they are not limited by the constraints of explicit functions, which directly express the dependent variable in terms of the independent variable.
An important aspect of implicitly defined functions is that they can describe one or more functions within a single equation. This is because the relationship between the variables may not be straightforward, and the equation might involve more complex interactions between the variables. The nature of these relationships could result in the graph of an implicitly defined function consisting of a single curve, multiple disjoint curves, or even an infinite number of curves.
To graph an implicitly defined function, we need to find solutions to the equation. A solution refers to the set of values for the variables that satisfy the equation. When we find solutions to an implicitly defined equation, we are essentially identifying the points on the graph where the relationship between the two variables holds true.
In order to create a graph of an implicitly defined function, start by identifying a range of values for one of the variables and then solving for the corresponding values of the other variable. The range should be based on the context of the problem or any constraints in the equation. Next, for each value in the chosen range, substitute it into the equation and solve for the other variable (y).
Once you have found the corresponding values for the other variable, plot these points on the xy-plane, which will help you visualize the relationship between the two variables as a curve or a set of curves. After plotting the points, examine the graph's characteristics, such as continuity, smoothness, or the presence of any asymptotes or singularities. These features can provide valuable insights into the underlying relationship between the variables.
Remember that an implicitly defined function can represent one or more functions. In some cases, solving the equation for one of the variables can help reveal an explicit function whose graph is part or all of the graph of the original implicitly defined function.
The graph of an implicitly defined function, x^{2} + y^{2} = 1 is shown. The graph represents a circle with a radius of 1 and centered at the origin (0,\, 0).
When working with implicitly defined functions, it is essential to recognize that they can represent one or more functions. In some cases, it may be possible to solve the equation for one of the variables to obtain an explicit function, whose graph would be part or all of the graph of the original implicitly defined function.
Examples
Example 1
Graph the implicitly defined function represented by the equation x^{2} + y^{2} = 4.
Implicitly defined functions are equations that involve two variables but do not explicitly state the dependent variable in terms of the independent variable. These functions can represent one or more functions within a single equation, as the relationship between the variables may be more complex than that of explicit functions.
The shape of the graph can be identified by analyzing the given equation and generating a set of ordered pairs that satisfy the equation.
In some cases, solving the equation for one of the variables can reveal an explicit function whose graph is part or all of the graph of the original implicitly defined function.
Variation in implicitly definied functions
An implicitly defined function is an equation that involves two variables, where the relationship between the variables is not explicitly given in the form of one variable being expressed as a function of the other. To understand how these two quantities vary together, we need to carefully examine the equation and observe the behavior of the variables as they change with respect to each other.
To analyze how the quantities in implicitly defined functions vary together, we can follow a more detailed process involving several steps. First, we need to carefully examine the equation and look for patterns or relationships between the terms involving both variables. Sometimes, rearranging the equation or partially solving for one variable can provide a clearer view of the relationship between the variables. The goal is to gain a better understanding of how changes in one variable might affect the other.
Next, we should compute the changes in both variables for a given interval. This calculation helps us determine whether the variables are increasing, decreasing, or remaining constant during that interval. Understanding how the variables change in relation to each other can provide valuable insights into the nature of their relationship.
After calculating the changes, we need to observe the signs of these changes. Examining the signs can help us infer the type of relationship between the variables. If the changes in both variables have the same sign (both positive or both negative), it indicates a positive relationship, meaning the variables increase or decrease together. On the other hand, if the changes have opposite signs (one positive and one negative), it suggests a negative relationship, implying that as one variable increases, the other decreases.
Lastly, we should look for instances where the rate of change of x with respect to y or of y with respect to x is zero. These intervals can reveal important information about the function's behavior, such as horizontal or vertical segments in the graph.
Here's an example of a visual description for the graph of an implicitly defined function.
x^{2} + y^{2} = 25
When analyzing the graph of an implicitly defined function, we can also observe intervals where the rate of change of x with respect to y or of y with respect to x is zero. These intervals can indicate vertical or horizontal segments of the graph.
| \text{Internal} | \text{Point A} | \text{Point B} | \Delta x | \Delta y | \dfrac{\Delta x}{\Delta y} | \text{Relationship} |
|---|---|---|---|---|---|---|
| \text{Interval 1} | (3,\,4) | (4,\,3) | 1 | -1 | -1 | \text{Negative} |
| \text{Interval 2} | (-3,\,4) | (-4,\,3) | -1 | -1 | 1 | \text{Positive} |
| \text{Interval 3} | (3,\,-4) | (4,\,-3) | 1 | 1 | 1 | \text{Positive} |
| \text{Interval 4} | (-3,\,-4) | (-4,\,-3) | -1 | 1 | -1 | \text{Negative} |
In this table, we show different intervals of the implicitly defined function (circle) with their respective points, the changes in x (\Delta x) and y (\Delta y), the ratio of the changes \dfrac{\Delta x}{\Delta y}, and whether the relationship is positive or negative. This table provides a clear representation of how the variables change with respect to each other and helps students understand the nature of their relationship in implicitly defined functions.
By understanding the relationship between the two variables in an implicitly defined function, we gain valuable insight into how the quantities vary together and can apply this knowledge to real-world situations and problem-solving.
Examples
Example 2
Determine how the quantities x and y vary together in the implicitly defined function given by the equation x^{2} + y^{2} - 6x - 8y = 9.
We analyzed the equation to understand the relationship between the variables x and y. We computed the changes in x and y for selected points close together on the graph and observed the signs of the changes in both variables to determine the nature of their relationship. Finally, we identified any intervals where the rate of change of x with respect to y or of y with respect to x is zero.