Unit 2 Graph-itti!

- Lesson 1 2.1 Symmetry (3-1)
- Lesson 2 2.2 Graph Families (3-2, 3-3)
- Lesson 3 2.3 Inverses (3-4)
- Lesson 4 2.4 Continuity (3-5)
- Lesson 5 2.5 Extrema (3-6)
- Lesson 6 2.6 Rational Functions (3-7)

Warm-up

Unit Two Graph-itti

- In this unit we will learn
- STANDARD 2.1 use algebraic tests to determine

symmetry in graphs, including even-odd tests

(3-1) - STANDARD 2.2 graph parent functions and

perform transformations to them (3-2, 3-3) - STANDARD 2.3 determine and graph inverses of

functions (3-4) - STANDARD 2.4 determine the continuity and end

behavior of functions (3-5) - STANDARD 2.5 use appropriate mathematical

terminology to describe the behavior of graphs

(3-6) - STANDARD 2.6 graph rational functions (3-7)

STANDARD 2.1 use algebraic tests to determine

symmetry in graphs, including even-odd tests

(3-1)

- In this lesson we will
- Discuss what symmetry is and the different types

that exist. - Learn to determine symmetry in graphs.
- Classify functions as even or odd.

What is Symmetry?

- Point Symmetry Symmetry about one point
- Figure will spin about the point and land on

itself in less than 360º.

Formal Definition

P

M

P

Symmetry to Origin

- This is the main point we look at for symmetry.
- Lets build some symmetry!

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Determining Symmetry with Respect to the Origin

Lets do a couple

Line Symmetry

Whats that mean?

U D

Lines We Are Interested In

- x-axis
- y-axis
- y x
- y -x

x-axis

y-axis

y x

y -x

Checking Mathematically

Homework

- HW 2.1 P 134 15 35 odd

Warm-up

- Get a piece of graph paper and a calculator.
- Graph the following on separate axii

Homework

STANDARD 2.2 graph parent functions and perform

transformations to them (3-2)

- In this section we will
- Identify the graphs of some simple functions.
- Recognize and perform transformations of simple

graphs. - Sketch graphs of related functions.

Families of Graphs

- Any function based on a simple function will have

the basic look of that family. - Multiplying, dividing, adding or subtracting from

the function may move it, shrink it or stretch it

but wont change its basic shape.

Example

Reflections

Vertical Translations

Horizontal Translations

Vertical Dilations

Horizontal Dilations

Lets do some

Send One person from your group to get a white

board with a graph on it, a pen and an eraser.

STANDARD 2.2 graph parent functions and perform

transformations to them (3-3)

- In this section we will
- Use function families to graph inequalities.

So how do I graph this?

Homework

- HW1 2.2 P 143 13-29 odd, 33
- HW2 2.2 P 150 21-31 odd

Warm-up

Homework

STANDARD 2.3 determine and graph inverses of

functions (3-4)

- In this section we will
- Determine inverses of relations and functions.
- Graph functions and their inverses.

Inverse Relations

- An inverse of function will take the answers

(range) from the function and give back the

original domain.

Finding the inverse of a relation

- Easy!!! Just switch the domain and range!
- Are they both functions?

Property of Inverse Functions

- If f(x) and f 1(x) are inverse functions, then
- In other words
- Two relations are inverse relations iff one

relation contains the element (b,a) whenever the

other relation contains (a,b). - Does this remind you of something?

Lets look at them on a graph

A function and its inverse

- Are reflections of each other over the line y x.

Is the inverse a function?

Quick and dirty test

- If the original function passes the HORIZONTAL

line test then the inverse will be a function. - Lets check our parent graphs.

Is the inverse a function?

Is this a function?

Proving Inverses

- If two functions are actually inverses then both

the composites of the functions will equal x. - You must prove BOTH true.

Check to see if the following are inverse

functions

The Handy, Dandy Build Your Own Inverse Kit

- Replace f(x) with y (it is just easier to look at

this way). - Switch the x and y in the equation.
- Resolve the equation for y.
- The result is the inverse.
- Now check!

Try this

Word Problem

- The fixed costs for manufacturing a particular

stereo system are 96,000, and the variable costs

are 80 per unit. - A. Write an equation that expresses the total

cost C(x) as a function of x given that x units

are manufactured.

- B. Determine the equation for the inverse process

and describe the real-world situation it models.

- C. Determine the number of units that can be

made for 144,000.

Homework

- HW 2.3 P 156 15 39 odd and 45

Warm-up

Homework

STANDARD 2.4 determine the continuity and end

behavior of functions (3-5)

- In this section we will
- Determine the continuity or discontinuity of a

function. - Identify the end behavior of functions.
- Determine whether a function is increasing or

decreasing on an interval.

Continuity

- A continuous functions graph can be drawn

without ever lifting up your pencil. - It has no holes or gaps.

Discontinuities

- Anything which disrupts the flow of the graph.
- What parent graphs do we have which demonstrate

discontinuous functions?

Types of discontinuities p 159

- Function is undefined at a value but, otherwise,

the graph matches up. - Graph has a hole.

Types of discontinuities p 159

- Graph stops at one y-value, then jumps to a

different y-value for the same x-value. - Common in piece-wise functions.

Types of discontinuities p 159

- A major disruption in the graph.
- As graph approaches the domain restriction, the

graph will shoot towards either positive or

negative infinity.

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Continuity at a single value

Continuous? If not, what type of discontinuity

exists?

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Continuity on an interval

- A function is continuous on an interval iff it is

continuous at each number in the interval.

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Increasing and decreasing functions

AhhhhHuh?

- Increasing means uphill left to right.
- Decreasing means downhill left to right.
- Constant means a flat or horizontal line left to

right.

Behavior over an interval

Graph these on your calculator

- P 166 26, 28, 30
- Determine the intervals where the functions are

increasing or decreasing. - Write the intervals in interval notation and in

in terms of x.

Answers

26.

28.

30.

End Behavior

- What will the function be doing at the outermost

reaches of its domain and range?

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Check these out!

Homework

- HW 2.4 P 166 13 31 odd, 39
- You will need a graphing calculator.

Check these out! Find discontinuities, intervals

inc/dec and end behavior

Homework

STANDARD 2.5 use appropriate mathematical

terminology to describe the behavior of graphs

(3-6)

- In this section we will
- Find the extrema of functions.
- Learn the difference between Absolute Extrema and

Relative Extrema. - Find the point of inflection of a functions

(if it exists).

Extrema Minimums and Maximums

- An Absolute Minimum of Maximum is the lowest or

highest value the range of the function can have. - The slope of the line drawn tangent to the min or

max will have a slope of zero. - That point is called a critical point for the

graph.

Find the Extrema and Describe End Behavior

Relative Minimums and Maximums

- These points are not the absolute highs or lows

for the function but they are the high or low

over a certain interval. - The slope of the line tangent to a relative min

or max is still zero so the point is a critical

point. - Minimums are said to be concave up and maximums

are concave down.

Find the extrema

Points of Inflection

- A point of inflection occurs when a graph changes

from one concavity to another. - The slope of the tangent line to this point is

undefined ( a vertical line). This point is also

considered a critical point. - You will learn to calculate the point of

inflection in calculus.

Determine whether there is a minimum, a maximum

or a point of inflection at the given point

Homework

- HW 2.5 P 177 13 29 every other odd and 34
- You will need a graphing calculator.

Describe the end behavior of the graph.

STANDARD 2.6 graph rational functions (3-7)

- In this section we will
- Graph Rational Functions
- Determine vertical, horizontal and slant

asymptotes

Rational Functions

- Have a variable in the denominator.
- The denominator restriction will have a profound

effect on the functions graph.

Vertical Asymptotes and Holes

- Caused by values which make the denominator 0.
- Also known as removable and non-removable

discontinuities.

Holes or Removable Discontinuities

Vertical Asymptotes

Building a function

Horizontal and Slant Asymptotes

- 3 cases possible
- Degree of numerator lt Degree of denominator H.A.

at y 0. - Degree of numerator Degree of denominator H.A.

is the ratio of the coefficients. - Degree of numerator gt Degree of denominator Do

long division to find the Slant Asymptote.

Examples

Homework

- HW 2.6 P 186 15 39 odd, 43