9.2.1B Parabolas, Asymptotes & Graphs

Standard 9.2.1 Essential Understandings

The concept of function is one of the most important ideas in school mathematics. Functions are a way to study the relationship among two or more variables and organize the world. The definition of a mathematical function developed over a thousand years. The beginning focus was on the covariation of two variables. Studying how two variables change together constitutes much of the middle and secondary curriculum. As the definition of function evolved the "exactly one" feature of the function gained importance. This feature is highlighted when students are asked to distinguish between relations that are functions and those that are not.

In elementary school, students begin to study functions by finding rules for various patterns. The rule for determining the number of toothpicks in a figure is a function relating the figure number (independent variable) with the number of toothpicks in that figure (dependent variable).

Students explain that the number of toothpicks is the figure number multiplied by five.

Thinking of functions as rules that generalize patterns is developed further in middle school where students learn that a proportional relationship between two variables exists if the rule relating the variables is written as . They also can see this relationship graphically as a line passing through the origin. At the end of middle school, students are able to distinguish between linear functions and functions that are not linear by inspecting the function represented by a table, graph, equation, or a real-world situation.

Students at the secondary level use their knowledge of how linear functions change and relate it to how exponential and quadratic functions change. Among the functions that students study in high school is the rate of change as shown in tables, graphs, and equations, one of the most important features that distinguish the different types. Parts of the curves of quadratic and exponential functions can appear to be similar on a graph, but the differences in the rates of change show that no quadratic or exponential will ever match at more than just a few discrete points. Quadratic functions and exponential functions can both increase at an increasing rate but these patterns of change are quite different.

Functions are tools that people use to model relationships between two variables. In middle school students use linear functions to model real-world situations. The function w(t) = 1.5t + 7 could represent to weights of baby girls when they are t months old. Students learn how special features of the function relate to the real-world situation. The y-intercept represents the weight of the baby girl at birth, while the slope represents the number of pounds that the baby gains each month. In addition, the domain of the function is determined by the constraints of the real-world setting. Theoretically, the domain of any linear function could be the set of real numbers, but practically the function does not really make sense for values of t that are less than 0 months or larger than 6 months.

In high school students are expected to identify theoretical and practical domains for a variety of functions. In particular, students in secondary school will focus on the special features of quadratic, exponential, and reciprocals of linear functions. These special features provide insights into the real-world situation being modeled. The vertex of a quadratic function could be used to find the maximum or minimum value of the dependent variable while the asymptote of an exponential function could be used to describe the long-term behavior of a situation. Students in high school need to make connections among equations, graphs, tables, and real-world situations of a variety of functions at the high school level. Each representation provides insights into how student make sense of these functions.

Once students are able to identify important features of a variety of functions then they move to seeing how functions can be transformed. Initially students see graphs of functions as a collection of individual points and then move to seeing the graph as an object itself made up of an infinite number of points. Students initially graph functions like f(x) = (x + 3)² and g(x) = x + 2 by calculating points to put in a table, then using these points to create a graph. The more points they create the better the graph looks. They then learn to identify the intercepts and, if applicable, the vertex. This process reinforces the notion of a graph as a collection of a finite set of points. Eventually students are asked to combine functions to create new functions (i.e. find h(x) = f(x) + g(x) or k(x) = g(f(x))). These operations and compositions of functions force students to think of the functions themselves as objects. Students need to view the graphs of functions as objects when they are asked to explain how the graph of a function f(x) compares with the graph of the function m(x) where m(x) = f(x) + 5 or the graph of p(x) where p(x) = f(x + 7). Students at the high school level need to see that a vertical translation is the sum of the original function, and a constant function and a horizontal translation is a composition of the original function with a linear function that has a slope of one; i.e., the graph of the function w(x - 11) is a horizontal translation of the graph of w(x) 11 units to the right.

In high school students gain experience distinguishing among different types of functions and sort them into families. Students learn to identify key features of each type using graphs, tables, equations, and real-world situations. The students then begin to perform operations using the functions themselves as inputs. The results of these operations allow students to create new types of functions that can be used to solve new problems.

All Standard Benchmarks

9.2.1.1
Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain.

9.2.1.2
Distinguish between functions and other relations defined symbolically, graphically or in tabular form.

9.2.1.3
Find the domain of a function defined symbolically, graphically or in a real-world context.

9.2.1.4
Obtain information and draw conclusions from graphs of functions and other relations.

9.2.1.5
Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f(x) = ax² + bx + c, in the form f(x) = a( x - h)² + k , or in factored form.

9.2.1.6
Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function.

9.2.1.7
Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.

9.2.1.8
Make qualitative statements about the rate of change of a function, based on its graph or table of values.

9.2.1.9
Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations.