Algebra II Notes Linear Relations and Functions Unit 02. Special Functions
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1 Algebra II Notes Linear Relations and Functions Unit 0 Big Idea Special Functions This lesson examines three special functions; piecewise function usuall written with two or more algebraic expressions, a step function which consists of a series of line segments that look like steps, and an absolute value function that has a V- shape and is made up of two line with opposite slopes. Objectives F.IF.B. Interpret functions that arise in applications in terms of the context - For a function that models a relationship between two quantities, interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; smmetries; end behavior; and periodicit. F-IF.C.7 Analze functions using different representations-graph functions expressed smbolicall and show ke features of the graph, b hand in simple cases and using technolog for more complicated cases. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Skills Write and graph piecewise, step, and absolute value functions. Vocabular Piecewise-defined function - A function that is written using two or more expressions. Piecewise-linear function - A function in which the equation for each interval is linear. Step function - A function whose graph is a series of line segments. Greatest integer function - A step function, written as f ( x) to x. x, where f( x) is the greatest integer less than or equal Absolute value function - A function written as x if x 0 f ( x) 0 if x 0, for all values of x. x if x 0 Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
2 Algebra II Notes Linear Relations and Functions Unit 0 Examples, Notes, and Exam Questions Piecewise Function: a function that is represented b a combination of equations, each corresponding to a part of the domain Evaluating a Piecewise Function: Ex: Find f and f for the function f x x x, 0. x x0 Step One: Determine which equation to use based upon the value of x. f : x 0, so we will use the first equation. f : x 0, so we will use the second equation. Step Two: Substitute the value of x into the appropriate equation. f 9 f 9 Graphing a Piecewise Function: Ex: Graph the function f x x, x x x Step One: Graph the function x for all values of x less than. There should be an open circle on the point,. Step Two: Graph the function x for all values of x greater than in the same coordinate plane. There should be a closed circle on the point,. The Greatest Integer Function: g x x For ever real number x, gx is the greatest integer less than or equal to x. The greatest integer function is an example of a step function. Note: All points seen on the graph to the right are solid points. The left endpoints of each segment are open circles. Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
3 Algebra II Notes Linear Relations and Functions Unit 0 Graphing the Absolute Value Function x x 0 Recall: f x x x x 0 Note: The vertex is at the origin. vertex General Form of the Equation of an Absolute Value Function: a x h k Activit: Discover the effects the variables a, h, and k have on the graph: Plot points or use our graphing calculator to graph the following. Make notes on how the graph changes from the graph of x.. x. x. x. x. x. x Graphing from the General Form a x h k Vertex: hk, Line of Smmetr: x h If a 0, the V-shaped graph opens up. If a 0, the V-shaped graph opens down. The graph is wider than x when a, and narrower than x when a. Ex: Graph the absolute value function x. Step One: Plot the vertex,. Step Two: Plot a couple of points to the right of the vertex 0, and,. Step Three: Use smmetr to plot the reflection of the points to the left of the vertex. Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
4 Algebra II Notes Linear Relations and Functions Unit 0 Ex: Write the equation of the graph shown. Step One: Find the vertex and substitute the coordinates in for a x 0 hk, in the general form. 0, hk, Step Two: To find a, choose another point on the graph. Substitute the coordinates into the general equation and solve for a. Point:, a a a Solution: x SAMPLE QUESTIONS Write the equation of the piecewise function shown in the graph. QOD: In a piecewise function, wh must one part of the graph have an open circle as an endpoint, and the other have a closed circle as an endpoint? Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
5 Algebra II Notes Linear Relations and Functions Unit 0 Sample CCSD Common Exam Practice Question(s):. Which graph represents the piecewise function below? x, if x f ( x), if x x, if x. Which function represents the graph below? A. x, if 0 x f( x) x, if 0 x Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
6 Algebra II Notes Linear Relations and Functions Unit 0 B. C. D. x, if x,,, f( x) x, if x 0,,,, if 0 x, if x f( x), if x, if x, if 0 x, if x f( x), if x, if x. Graph the absolute value function x. QOD: How does the value of a relate to the slopes of the pieces of an absolute value function to the left and right of the vertex? Sample SAT Question(s): Taken from College Board online practice problems. In the x-coordinate plane, how man points are a distance of units from the origin? (A) One (B) Two (C) Three (D) Four (E) More than four x 7 x What value of x satisfies both of the equation? Grid-In Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
7 Algebra II Notes Linear Relations and Functions Unit 0 Sample CCSD Common Exam Practice Question(s):. Which is the graph of x? A. B. C. D. Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page 7 of //0
8 Algebra II Notes Linear Relations and Functions Unit 0. Graph the function: a. c. x x b. d. x x Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
9 Algebra II Notes Linear Relations and Functions Unit 0. Graph. a. c. x x b. d. x x ANS: D Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page 9 of //0
10 x = 0 x = 0 x = x = - Algebra II Notes Linear Relations and Functions Unit 0. Graph the absolute-value function, f(x) = x. Label the axis of smmetr and the vertex. Find the intercepts, and give the domain and range. a. Axis of smmetr c. Axis of smmetr (0, ) (0, ) (, 0) x (, 0) x b. The domain is all real numbers. The range is d. The domain is all real numbers. The range is (0, ) Axis of smmetr (, 0) (, 0) x x Axis of smmetr (0, ) The domain is all real numbers. The range is The domain is all real numbers. The range is. ANS: A Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
11 Algebra II Notes Linear Relations and Functions Unit 0. Graph. Compare with the graph of.\ a. The graph of is units above the graph of f(x) = x. c. The graph of is units to the left of the graph of f(x) = x. x x b. The graph of is units to the right of the graph of f(x) = x. d. The graph of is units below the graph of f(x) = x. x x ANS: B Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
12 Shipping Cost ($) Shipping Cost ($) Shipping Cost ($) Shipping Cost ($) Algebra II Notes Linear Relations and Functions Unit 0. A shipping compan charges $.0 to ship a package weighing one pound or less. Then the charge $.0 for each additional pound, or fraction of a pound, up to five pounds. Write a piecewise function that gives the price P for shipping a package weighing w pounds. Graph the function. a. c. Priorit Mail Rates Priorit Mail Rates Weight (lbs) b. d. Weight (lbs) Priorit Mail Rates Priorit Mail Rates Weight (lbs) Weight (lbs) ANS: C Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
13 Algebra II Notes Linear Relations and Functions Unit 0 7. Graph the function defined b. a. c. x x b. d. x x ANS: A. Graph the function. Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
14 Algebra II Notes Linear Relations and Functions Unit 0 ANS: Alg II Notes Unit 0g Special Functions (Piecewise, Absolut Value, etc) Page of //0
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