CHAPTER 1 FUNCTIONS AND GRAPHS

Quick Talk What can you tell me about functions based on your peers’ lessons? What can you tell me about graphs based on your peers’ lessons?

Review about factoring

Challenge: Could you factor these?

Quick Talk: What does factoring allow you to find?

Potential Answers Rational Zeros when y=0

A mathematical model is a mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. There are 3 types of model: Numerical Models Algebraic model Graphical models

The most basic type: Numerical Model Numerical Model: use of numbers or data are analyzed to gain insights into phenomena

Group work: Numerical Model Example #1 YearTotalMaleFemale The chart shows growth in the number of engineers from Is the proportion of female engineers over the years increasing? Explain

Answer You have to be careful with the last example, the word “proportion” means what? Once we converted all the data into ratios, we see that there is an increase from , but there is a drop from , then from it increased again. From the data, the peak is at 1990, 32.3% YearTotalFemaleActual ratio

Algebraic Model Algebraic model uses formulas or equations to relate variable quantities associated with the phenomena being studied.

Group Work: Algebraic Model Example #1 Note: Think about what are you comparing. What “formula” can you use? Or do you have to create one?

Answer

Group Work: Algebraic Model Example #2 You went to Gamestop, it just happens every game you buy is discounted 15%( ) off the marked price. The discount is taken at the sales counter, and then a state sales tax of 7.5% and local tax of 1% is added on. Questions: 1) If you have $20, could you buy a game marked at $24.99? 2) If you are determined to spend no more than $175, what’s the maximum total value of your marked purchases be?

Answer 1) Identify your variables (ex: let m=market price, d=discounted price, k=constant, t=taxes, s=total sale price) Your general equation should be: d=km s=d+td When you substitute: s=km+t(km) Now determine the values for each variable: k=.85 m=? d=? t=.085 s=? Your actual equation (with numbers) should be: d=.85m s=.85m+.085(m) 1) m=24.99, s=.85(24.99)+.085(24.99), s= , s= Since you only have $20, 20 < 23.36, therefore, you can not buy the game 2) s=175, so 175=.85(m)+.085(m), m=187.17, so the maximum total value of the market price should be at most $187.17

Graphing Model Graphing model is a visible representation of a numerical model or an algebraic model that gives insight into the relationships between variable quantities.

Graphical model Example #1 Graph this. What does it look like? Can you find an algebraic model that fits?

Answer

Graphical Model example #2 Time (t) Females (f) Create a graphic model, that fits the best with these data sets.

Answer

Group work Situation:

Answer

Group Work Solve for this equation

Answer X=0 or x=5/2 or x=-2/3

Group Work Solve this

Homework Practice Pgs #1-9odd, 11-18, 29, 31, 35,43, 45, 48, 50

BASIC PARENT FUNCTIONS

Quick Talk: What makes a function? What does it consists of?

Answer One to one / passes vertical line test Y and x Equation Domain Range Note: y is aka f(x)

Quick Talk: From what you just learned, which of these is not a function?

Quick Talk: What are parent functions?

Parent Function is the simplest function with characteristics. It is without any “transformations” or “shifting”

12 parent functions: As you are graphing it, please draw an arrow at the end.

The Identity Function Slide 1- 35

The Squaring Function Slide 1- 36

The Cubing Function Slide 1- 37

The Reciprocal Function Slide 1- 38

The Square Root Function Slide 1- 39

The Exponential Function Slide 1- 40

The Natural Logarithm Function Slide 1- 41

The Sine Function Slide 1- 42

The Cosine Function Slide 1- 43

The Absolute Value Function Slide 1- 44

The Greatest Integer Function Slide 1- 45

The Logistic Function Slide 1- 46

Important!!! Read this!!! I neeeeeeedddddddd you guys to recognize, understand and know the parent functions!!!!! Why? Because it will make finding domain and range easier.

What “shifts” can you possibly have on a parent function? You can shift a function left or right (horizontal shifts) You can shift a function up or down (vertical shifts) Function can be steeper or flatter (linear) Function can be wider or narrower (horizontal shrink/vertical stretch or horizontal stretch/vertical shrink) Function can flip across x or y axis

Group Activity: graphing! Each group will select a member to put it on the whiteboard. With t chart. X:[-5,5]

Shifts review

Name the parent function then the shifts

Answer Shift right 5 Shift down 9 Vertical stretch of 3/2 or horizontal shrink of 2/3

Homework Practice P147 #1,4,5,7,9,17,22

Domain and Range Domain are all the input values of the function. Range are all the output of the function.

Looking at Domain and Range Graphically What is the parent function? What is the domain? What is the range?

Answer

Group Work: What is the domain and range?

Answer Domain: [-5,5] Range: [-3,3]

Group Activity: Find the Domain and Range of the 12 parent functions. Note: The answers are not on my slides, I will be showing my answers right now in class. If you miss this, get it from me or another person from your class.

As you can see from the step function, not every domain or range are continuous.

Slide Example Identifying Points of Discontinuity Which of the following figures shows functions that are discontinuous at x = 2?

Slide Continuity

Finding Domain and range Algebraically

How do you find the domain algebraically?

Group Work: Determining the domain and range

Look at our example #1 and #3 We have something unique. It is called an asymptote.

Slide Horizontal and Vertical Asymptotes

Asymptotes Remember the graph will get infinitely close to the asymptotes but will NEVER intersect with it.

Slide Increasing, Decreasing, and Constant Function on an Interval A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x). A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x).

Slide Increasing and Decreasing Functions

Group work: Find the Increase and Decrease

Shown in class: How do you graph it?

Slide Answer:

Slide Local and Absolute Extrema A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f. A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f. Local extrema are also called relative extrema.

In another word You may have multiple relative/local maximum and relative/local minimum. They are located where the slopes are = 0 (important concept in calculus) Absolute “maximum” or “minimum” is where it is the most “top” or “bottom point” of the function.

Slide Lower Bound, Upper Bound and Bounded A function f is bounded below of there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. A function f is bounded above of there is some number B that is greater than or equal to every number in the range of f. Any such number B is called a upper bound of f. A function f is bounded if it is bounded both above and below.

In another word, Function is bounded below, if there is an “absolute minimum” or if there is a “floor” Function is bounded above, if there is an “absolute maximum” or if there is a “ceiling”

End Behavior Asymptote

Example: Find end behavior asymptote

Answer:

Group work: Find end behavior asymptote

Answer:

Symmetry Symmetry: a line where you can fold one side onto the other.

Slide Symmetry with respect to the y-axis

Slide Symmetry with respect to the x-axis

Slide Symmetry with respect to the origin

Slide Example Checking Functions for Symmetry

Slide Example Checking Functions for Symmetry

Group work: Look at the 12 parent functions. Find the boundedness, location of local/absolute min/max, where it is increasing, decreasing or constant.

Group work:

Answer shown in class

Homework Practice

BUILDING FUNCTIONS FROM FUNCTIONS

Overview It is important how you can put functions together.

Slide Sum, Difference, Product, and Quotient

Slide Example Defining New Functions Algebraically

Group work: From the last examples, find the following: Domain: Range: Continuous: Increase/decrease: Symmetric: Boundedness: Max/min: Asymptotes: End behavior:

Slide Composition of Functions

Note: Composition is like “function within a function”

Composition Examples

Answer:

Group Work:

Answer shown in class

Group Work:

Answer

Working backward

Possible Answers

Group Work:

Answer

Word problem A high-altitude spherical weather balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.03 inches per second and that r=48 inches at time t=0. Determine an equation that models the volume V of the balloon at time t and find the volume when t=300 seconds.

Answer

Word Problem #2 A satellite camera takes a rectangular shaped picture. The smallest region that can be photographed is a 5km by 7km rectangle. As the camera zooms out, the length l and width w of the rectangle increase at a rate of 2 km/s. How long does it take for the area A to be at least 5 times its original size?

Answer

Implicitly: Something not directly expressed

Group Work: (try factor it!)

Answer

Homework Practice P124 #2,4,6,7,12,16,19,23,24,34,37,44

PARAMETRIC RELATIONS AND INVERSES

Parametric Equation A function that define both elements of the ordered pair (x,y) in terms of another variable t.

Example of Parametric

Answer

Group Work: Graph Parametric Equation

Answer t(x,y) -3(3,-2) -2(0,-1) (-1,0) 0(0,1) 1(3,2) 2(8,3) 3(15,4)

How to do parametric equation with the calculator.

Slide Inverse Relation The ordered pair (a,b) is in a relation if and only if the pair (b,a) is in the inverse relation.

Slide Horizontal Line Test The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

Slide Inverse Function

How do you find the inverse?

Group work: Find the inverse

Slide Group Work:

Slide Answer

Slide The Inverse Reflection Principle The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y=x. The points (a,b) and (b,a) are reflections of each other across the line y=x.

Slide The Inverse Composition Rule

Group Work

Homework Practice P135 #2,5,9-12, 14,16,22,27,29,33

MODELING WITH FUNCTIONS Please bring your book

Quick Talk: Name the following formulas 1) Volume of Sphere 2) Volume of Cone 3) Volume of Cylinder 4) Surface Area of Sphere 5) Surface Area of Cone 6) Surface Area of Cylinder 7) Area of circle 8) Circumference

Answer

Slide Example A Maximum Value Problem

Answer

Group Work Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

Answer

I hope you bring your book Classwork P160 #1-35 every other odd