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Warm Up • Write down objective and homework in agenda • Lay out homework (Function Worksheet) • Homework (Function Notation Worksheet) Unit 3 Functions -Common Core Standards • • • • • • 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Note: At this level, focus on linear and exponential equations F-BF.1 Write a function that describes a relationship between two quantities. 1)Determine an explicit expression, a recursive process, or steps for calculation from a context. 2) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to linear or quadratic functions. F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not addressed. F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Unit 3 Functions -Common Core Standards • • • • • • F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: At this level, the focus is linear and exponential functions. F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Note: At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity. F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Note: At this level, focus on linear and exponential functions F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: At this level, focus on linear functions and exponential functions whose domain is the subset of integers. N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Warm UP • List the domain and range for all but #3 Functional Notation • An equation that is a function may be expressed using functional notation. • The notation f(x) (read “f of (x)”) represents the variable y. Functional Notation Example: y = 2x + 6 can be written as f(x) = 2x + 6. Given the equation y = 2x + 6, evaluate when x = 3. y = 2(3) + 6 y = 12 Functional Notation For the function f(x) = 2x + 6, the notation f(3) means that the variable x is replaced with the value of 3. f(x) = 2x + 6 f(3) = 2(3) + 6 f(3) = 12 Evaluating Functions Given f(x) = 4x + 8, find each: 1. f(2) = 4(2) + 8 = 16 2. f(a +1) = 4(a + 1) + 8 = 4a + 4 + 8 = 4a + 12 3. f(4a) = 4(-4a) + 8 = -16a+ 8 Given f(x) = 3x - 2, find: 1) f(3) = 7 3 3(3)-2 7 3(-2)-2 -8 2) f(-2) = -8 -2 Given h(z) = z2 - 4z + 9, find h(-3) -3 2 (-3) -4(-3)+9 9 + 12 + 9 h(-3) = 30 30 Evaluating More Functions If f(x) = 3x 1, and g(x) = 5x + 3, find each: 1. f(2) + g(3) = [3(2) -1] + [5(3) + 3] = 6 - 1 + 15 + 3 = 23 2. f(4) - g(-2) = [3(4) - 1] - [5(-2) + 3] = 11 - (-7) = 18 3. 3f(1) + 2g(2) = 3[3(1) - 1] + 2[5(2) + 3] = 6 + 26 = 32 Given g(x) = 1. 2. 3. 4. 2 6 14 18 2 x – 2, find g(4) Given f(x) = 2x + 1, find -4[f(3) – f(1)] 1. 2. 3. 4. -40 -16 -8 4 Use the graph to answer the questions • What is the domain? • What is the range? f(x) 26 24 20 16 12 8 4 x 0 4 8 12 16 20 24 26 Use the graph to answer the questions • What is the domain? 0 ≤ x ≤ 26 • What is the range? 0 ≤ y ≤ 20 5 16 + 5 = 21 3 x 4 = 12 10/2 = 5 20 – 8 = 12 Functions on a Calculator! • To evaluate functions on your calculator, enter the function into y=. In this case, enter 19.4+1.28X into Y1. • Your calculator uses the notation Y1(X) instead of f (x). To enter a statement using function notation, go to the home screen. • To calculate the value for above, 𝑓(−10), we must enter Y1(-10). To find Y1, hit VARS, arrow right to Y-VARS, select option 1: Function, and then select option 1: Y1. • Back at the home screen, complete the calculation by typing (-10) and hitting ENTER. Challenge!!!!!!!!!! • What does it mean when we have this.. g(f(x)) GREAT Practice! • http://www.mathplayground.com/functionm achine.html • http://nlvm.usu.edu/en/nav/frames_asid_19 1_g_4_t_2.html • http://regentsprep.org/Regents/math/ALGE BRA/AP3/FuncPrac.htm • http://www.algebra-class.com/vertical-linetest-practice.html