Functions & Non-Linear Graphs
Unit 7 - Functions & Non-Linear Graphs
Learning Targets/Performance Indicators
- Understand the concept of a function as assigning to each element of the domain exactly one element of the range.
- Use function notation and interpret statements that use function notation, including notation in terms of a given situation.
- Determine the domain and range of a given function (both algebraically and graphically) and a function in a given context.
- Evaluate functions for any input.
- Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- Interpret key features of graphs and tables of functions (i.e., intercepts, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums, and symmetries).
- Calculate and interpret the average rate of change of a function over a specified interval.
- Estimate the rate of change from a graph.
- Graph functions showing intercepts, maxima, and minima, and end behavior.
- Graph absolute value, step, and piecewise defined functions.
- Utilize technology to graph complicated functions.
- Utilize technology to determine where the graphs of two functions intersect.
- How do you determine the domain and the range of a function and in what situations could there be restrictions on the domain and the range?
- How can you determine if a relation is a function?
- What are the key features of a function and how could you use them to sketch a graph of the function?
- What is the advantage of using the notation f(x) = as opposed to y = ?
- How would you represent a sequence using function notation?
- How do you graph various types of functions (linear, exponential, quadratic, absolute value, step, piece-wise defined), and what are the key features of the graphs of these functions?
- What are the various transformations on the graph of a parent function and how do those change the parent function?
- How is technology used to make connections between graphs, tables of values and solutions?
- How can expressions be rewritten in equivalent forms and how is this useful?
- In what situations would it be appropriate use a linear model verses an exponential model?
In grade 5, students defined the coordinate system and graphed ordered pairs called coordinates. (5.G.1) In grade 6, students drew polygons in the coordinate plane given coordinates for the vertices. (6.G.3) In grade 7, students drew (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. (7.G.2) Students also solved problems involving scale drawings of geometric figures. This included computing actual lengths and areas and reproducing scale drawings at a different scale. (7.G.1) In grade 8, students verified experimentally the properties of rotations, reflections, and translations. They learned that lines, segments, and angles maintain their shape and size when transformed. (8.G.1) Students recognized that a two-dimensional figure is congruent (8.G.2)/similar (8.G.4) to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Furthermore, students described the effect of dilations, translation, rotations, and reflections on two-dimensional figures using coordinates. (8.G.3)
Also in grade 8, students used functions to model relationships between quantities and constructed a function to model a linear relationship between two quantities. They determined the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading quantities by analyzing a graph. (8.F.4 and 8.F.5) In the previous unit of this course, students were introduced to function notation. They evaluated and interpreted functions, and determined the domain and range of a function.
Students graph linear and quadratic functions and show intercepts, maxima, and minima. Students graph exponential functions, absolute value functions, step functions, and piecewise-defined functions. Technology is used to graph more complicated cases and to also determine where the graphs of two functions intersect. Students determine an explicit expression, a recursive process or the steps for calculation from a given situation in order to write a function that describes a relationship between two quantities. They build functions to model situations by combining standard function types using arithmetic operations.
In algebra 2 and precalculus, students will use their knowledge of transformations when building functions from existing functions. Students pursuing art-related programs and careers will continue the study of visual transformations. Students will graph and identify key features of the parent functions f(x) = x2, f(x) = |x|, f(x) = √x, and f(x) = x3. In addition, students will continue to explore transformations of selected functions. They will also graph more advanced piecewise and step functions. (F.IF.4, F.IF.5, F.IF.7)