Can inverse variation be linear

Introduction to Linear Functions. Concept Version 4. Learning Objective Relate the concept of slope to the concepts of direct and inverse variation. Key Points Two variables that change proportionally to one another are said to be in direct variation. The relationship between two directly proportionate variable can be represented by a linear equation in slope -intercept form , and is easily modeled using a linear graph.

Inverse variation is the opposite of direct variation; two variables are said to be inversely proportional when a change is performed on one variable and the opposite happens to the other. The relationship between two inversely proportionate variables cannot be represented by a linear equation, and its graphical representation is not a line, but a hyperbola.

Full Text Direct Variation Simply put, two variables are in direct variation when the same thing that happens to one variable happens to the other. Inversely Proportional Function An inversely proportional relationship between two variables is represented graphically by a hyperbola.

Edit this content. Prev Concept Slope. Zeroes of Linear Functions. We can then solve for k by using the given values for x and y. To find the value of y , we substitute for x. Using a Ratio to Model Direct Variation. The direct variation model can be rewritten in ratio form:.

The ratio form lets you know that when x and y have direct variation, the ratio of x to y is alike for all values of x and y. Occasionally, real-life figures can be approximated by a direct variation model, although the data might not fit the model exactly.

The lengths in centimeters of 8 Maltese dogs at age 1 and age 10 are shown in the table below. Write a model that relates the length of the dogs at 1 year old to the length at 10 years old. Estimate the body length of a Maltese whose length at 1 year was We need to start by finding the ratio of 1 year to 10 years for each Maltese.

In a classroom focused on symbolic manipulation, students may develop an implicit belief that variables are merely placeholders for numbers. Consequently, the variable does not vary. Direct variation appears throughout the Algebra 1 curriculum. Inverse variation: When the ratio of one variable to the reciprocal of the other is constant i.

Alternative definition: One quantity is inversely proportional to another when the product of the two quantities is constant. The equation of an inverse proportion can also be written in the form. Inverse variation provides a rich curricular complement to direct variation. Direct variation and inverse variation are related topics, and it makes sense to study them in parallel. Because they have striking differences, the contrast allows students to gain a deeper understanding of various functions.

Inverse variation allows students to consider nonlinear functions. The graph of an inverse variation never crosses the x-axis or the y-axis, nor does it pass through the origin. When teaching inverse variation — as with direct variation and other activities involving mathematical modeling — asking students to gather data helps to spark their interest.

Making sound mathematical decisions is the basis of effective modeling, so providing opportunities for students to make choices helps to develop their analytical abilities. For real-world explorations involving inverse variation, it will be necessary to collect enough data to make the nonlinear pattern obvious. Too few points may result in a pattern that appears to be linear. Once sufficient data have been collected, students can use tables and graphs to represent the data.

Finally, students should compare direct variation with indirect variation, illuminating the differences and highlighting the similarities. They should recognize that the constant of proportionality in the direct variation is a quotient of the variables, while the constant of proportionality in the inverse variation is a product. Or they might consider the graphs, since a direct variation is linear and passes through the origin, while an inverse variation is a curve with no x- or y-intercepts.

Making these comparisons will allow students to understand the differences within a family of functions. The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Effective Classroom Questioning This brochure from the Center for Teaching Excellence at the University of Illinois at Urbana-Champaign provides a brief overview of questioning techniques, with tips on how to ask questions effectively in the classroom.

Brualdi, discusses good questions, bad questions, and how to ask questions that foster student achievement. Environmental Protection Agency. In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations.

In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations. In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them.