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Week # 7 Lecture – pp 78-104 Lecture Presentations for Integrated Biology and Skills for Success in Science Banks, Montoya, Johns, & Eveslage

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Lecture Week 7— Functions, Processes and Non-Linear Equations By the end of the lecture, students will be able to: 1. Determine if an equation is a function or not. 2. Identify the which functions are able to be inverted and which are not. 3. Find the inverse of a function, when one exists. 4. Determine the input when given the output, and vice versa. 5. Use function notation to solve problems. 6. Graph non-linear equations (i.e., quadratic, cubic, exponential, piece-wise and step). 7. Determine symmetries on a graph.

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Functions A function is a process that will have exactly one output for every input. This means that you cannot put 5 into the function machine one time and get 10, and then put 5 in again and get something different than 10—you must always get the same output for a given input. The function notation is written as f(x), which means that you take the input of “x” and perform the function on it. This is said “f of x”

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Is this a function? When x = 2, y can equal -2 or 4... therefore, it’s NOT A FUNCTION

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Functions (Cont.) Example:y = 3x + 2 slope/intercept form f(x) = 3x + 2 function notation Find f(4). This is the same problem as “find y when x is 4.” Said “f of 4” x=4 y = 3x + 2f(x) = 3x + 2 y = 3(4) + 2f(4) = 3(4) + 2 y = 12 + 2f(4) = 12 + 2 y = 14f(4) = 14 This is function notation

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Process Diagrams One way to visual represent a function is a process diagram. Using the function f(x) = 2x + 6 here’s what a process diagram would look like: x multiply by 2 2x add 6 2x+6 = y You start with x, the input, and get y, the output. The operations go inside the boxes. Try to make a process diagram for: g(x) = x – 5

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Inverse Processes Sometimes, processes can be inverted. This is not the same as the opposite, and should only be referred to as the inverse. Remember the process for f(x) = 2x + 6 x multiply by 2 2x add 6 2x+6 = y Try to make a process diagram that would UNDO the process for f(x). (Hint: go backwards and do the inverse of each box.) x subtract 6 x – 6 divide by 2 x – 6 = y 2

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Invertible Processes (Continued) What processes in science have you learned about are invertible? Think about making a monomer into a polymer. H-monomer-OH + H-monomer-OH +... What was this process called? Why? Can this process be UNDONE? (Is it invertible?) What is the name of the inverse process?

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Non-linear

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Quadratic Functions

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Graphing Quadratics

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Quadratic Functions

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Step Functions The United Postal Service charges $2 per pound to ship a package. Any value in between pounds is rounded down. Graph this function. (Your graph should look like a stair step.)

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Piece-wise functions Graph this function on a distance vs. time graph. For the first four seconds you walk at 3 m/s. Then you slow down to 2 m/s for seconds 4-10. Then you run as fast as you can for seconds 10-20 at a rate of 6 m/s, and then you stop. Graph this data. Start with a table—be sure to put every point on your table where there is a change in the slope (rate).

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Exit Quiz and Homework

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EQ: What are the characteristics of functions and their inverses?

EQ: What are the characteristics of functions and their inverses?

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