Presentation on theme: "Week # 7 Lecture – pp 78-104 Lecture Presentations for Integrated Biology and Skills for Success in Science Banks, Montoya, Johns, & Eveslage."— Presentation transcript:
Week # 7 Lecture – pp 78-104 Lecture Presentations for Integrated Biology and Skills for Success in Science Banks, Montoya, Johns, & Eveslage
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.
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”
Is this a function? When x = 2, y can equal -2 or 4... therefore, it’s NOT A FUNCTION
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
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
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
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?
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.)
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).