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.

FunctionsA 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=4y = 3x + 2 f(x) = 3x + 2

y = 3(4) + 2 f(4) = 3(4) + 2y = 12 + 2 f(4) = 12 + 2

y = 14 f(4) = 14

This is function notation

Process DiagramsOne 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 ProcessesSometimes, 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?

Non-linearA non-linear equation is any equation

that does not form a straight line when graphed.

Here are some examples:y = (a quadratic function)y = (a cubic function)y = (an exponential function)A piece-wise functionA step function

No matter what type of function it is, you can always start graphing with a table.

Quadratic FunctionsQuadratic Functions have the general form:

f(x) = The highest exponent on an x is a 2. This defines it as a quadratic function. Quadratic comes form the word quadratum, meaning square.

The coefficients (a, b, and c) are sometimes tricky to determine. What are the coefficients for this equation?

y =

a = 1 b = 0 c = -2

Graphing QuadraticsHere’s the graph of y =

What would the graph of y = ) look like?

Remember that the negative is applied AFTER the exponent (Parenthesis, Exponent, Multiplication/Division, Addition/Subtraction)

Quadratic FunctionsUse the same graph paper to graph these functions. (Hint: make a table for each one and use the x-values of -3, -2, -1, 0, 1, 2, and 3.)Graph y = Graph y = Graph y = Graph y = Graph y =

State a rule for when a number is added before the “squared” step and a rule for when a number is added after the “squared” step.

Step FunctionsThe 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 functionsGraph 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).

Exit Quiz and HomeworkExit Quiz—Copy the questions, then answer.1. Consider the function: f(x) = -2x – 7 Find f(x) when x = -3 Find x when f(x) = 11 Write a process diagram for f(x). Write the inverse of f(x) as g(x).2. Graph y = + 2 and y = + 2 on separate

graphs. Find and mark any lines of symmetry.

HomeworkRead and annotate Chapters 10 and 11.Study for EXAM.Review your notes, the syllabus and course objectives from class. (Be sure you understand the objectives.)