FUNCTIONS AND GRAPHS. Aim #1.2 : What are the basics of functions and their graphs? Let’s Review:...

FUNCTIONS AND GRAPHS

Aim #1.2: What are the basics of functions and their graphs?

Let’s Review: What is the Cartesian Plane or Rectangular

Coordinate Plans? How do we find the x and y-intercepts of any

function? How do we interpret the viewing rectangle

[-10,10, 1] by [-10, 10,1]?

IN THIS SECTION WE WILL LEARN:

How to find the domain and range? Determine whether a relation is a function Determine whether an equation represents a

function Evaluate a function Graph functions by plotting points Use the vertical line to identify functions

WHAT IS A RELATION?

A relation is a set of ordered pairs. Example: (4,-2), (1, 2), (0, 1), (-2, 2)

Domain is the first number in the ordered pair. Example:(4,-2)

Range is the second number in the ordered pair.

Example: : (4,30)

EXAMPLE 1:

Find the domain and range of the relation.

{(Smith, 1.0006%), (Johnson, 0.810%), (Williams, 0.699%), (Brown, 0.621%)}

PRACTICE:

Find the domain and range of the following relation.

HOW DO WE DETERMINE IF A RELATION IS A FUNCTION?

A relation is a function if each domain only has ONE range value.

There are two ways to visually demonstrate if a relation is a function.

1. Mapping2. Vertical Line Test

DETERMINE WHETHER THE RELATION IS A FUNCTION

a. {(1, 6), (2, 6), (3, 8), (4, 9)}b. {(6, 1), (6, 2), (8, 3), (9, 4)}

FUNCTIONS AS EQUATIONS

Functions are usually given in terms of equations instead of ordered pairs.

Example: y =0.13x2 -0.21x + 8.7

The variable x is known the independent variable and y is the dependent variable.

HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A

FUNCTION?

x2 + y = 4 Steps: Solve the equation

for y in terms of x. Note: If two or more y

values are found then the equation is not a function.

HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A FUNCTION

x2 + y2 =4 Steps: Solve the equation

for y in terms of x. Note: If two or more y

values are found then the equation is not a function.

PRACTICE:

Solve each equation for y and then determine whether the equation defines y as a function of x.

1. 2x + y = 62. x2 + y2 = 1

WHAT IS FUNCTION NOTATION?

We use the special notation f(x) which reads as f of x and represents the function at the number x.

Example: f (x) = 0.13x2 -0.21x +8.7 If we are interested in finding f (30), we

substitute in 30 for x to find the function at 30.

f (30)= 0.13(30)2 -0.21 (30) + 8.7 Now let’s try to evaluate using our

calculators.

HOW DO WE EVALUATE A FUNCTION?

F (x) = x2 + 3x + 5

Evaluate each of the following:

f (2) f (x + 3) f (-x)

Substitute the 2 for x and evaluate.

Then repeat.

GRAPHS OF FUNCTIONS

The graph of a function is the graph of the ordered pairs.

Let’s graph: a. f (x) = 2xb. g (x) = 2x + 4

USING THE VERTICAL LINE TEST

The Vertical Line Test for Functions If any vertical line intersects a graph in more

than one point, the graph does not define y as a function of x.

Practice:

Use the vertical line test to identify graphs in which y is a function of x.

SUMMARY: ANSWER IN COMPLETE

SENTENCES.

What is a relation? What is a function? How can you determine if a relation is a

function? How can you determine if an equation in x

and y defines y as a function of x? Give an example.

AIM #1.2B: WHAT KIND OF INFORMATION CAN WE OBTAIN FROM GRAPHS OF FUNCTIONS?

Note the closed dot indicates the graph does not extend from this point and its part of the graph.

Open dot indicates that the point does not extend and the point is not part of the graph.

HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A FUNCTION’S GRAPH?

Domain: set of inputs

Found on x –axis

Range: set of outputs Found on y -axis

Using set builder notation it would look like this for the domain:

Using Interval Notation: [-4, 2]

What would it look like for the range using both set builder and interval notation?

}24{ xx

IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH

Use Set Builder Notation.

Domain:

Range:

IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM ITS GRAPH

IDENTIFYING INTERCEPTS FROM A FUNCTION’S GRAPH

We can say that -2, 3, and 5 are the zeros of the function. The zeros of the function are the

x- values that make f (x) = 0.Therefore, the real zeros

are the x-intercepts.

A function can have more than one x-intercept, but at most one y-intercept.

SUMMARY: ANSWER IN COMPLETE

SENTENCES.

Explain how the vertical line test is used to determine whether a graph is a function.

Explain how to determine the domain and range of a function from its graph.

Does it make sense? Explain your reasoning.

I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.

AIM #1.3: HOW DO WE IDENTIFY INTERVALS ON WHICH A FUNCTION IS

INCREASING OR DECREASING?

Increasing, Decreasing and

Constant Functions

A function is increasing on a open

interval, I if f (x1) < f(x2)

whenever x1<x2 for any x1 and x2 in the

interval.

A function is decreasing on an open interval, I, if f(x1) > f (x2) whenever x1 > x2

for any x1 and x2 in the interval.

A function is constant on an open interval, I, f(x1) = f (x2) for any x1 and x2 in the interval.

Note: The open intervals describing where function

increase, decrease or are constant use x-coordinates and not y-coordinates.

Example 1: Increases, Decreases or Constant

State the interval where the function is increasing, decreasing or constant.

Practice:

State the interval where the function is increasing, decreasing or constant.

WHAT IS A RELATIVE MAXIMA?

Definition of a Relative Maximum

A function value f (a) is a relative maximum of f if there exists an open interval containing a such that f (a) > f (x) for all x ≠ a in the open interval.

WHAT IS A RELATIVE MINIMA?

Definition of a Relative Minimum A function value f

(b) is a relative minimum of f if

there exists an open interval containing b

such that f (b) < f (x) for all x ≠ b in the open interval.

HOW DO WE IDENTIFY EVEN AND ODD FUNCTIONS AND SYMMETRY? Definition of Even and Odd Functions

The function f is an even function if f (-x)= f (x) all x in the domain of f.

The right side of the equation of an even function does not change if x is replaced with

–x.

The function f is an odd function if f (-x) = -f (x) for all x in the domain of f.

Every term on the right side of the equation of an odd function changes its sign if x is

replaced with –x.

DETERMINE IF FUNCTION IS EVEN,ODD OR NEITHER

f (x) = x3 - 6x Steps: Replace x with –x

and simplify. If the right side of

the equation stays the same it is an even function.

If every term on the right side changes sign, then the function is odd.

DETERMINE IF FUNCTION IS EVEN, ODD OR NEITHER

g (x) = x4 - 2x2

h(x) = x2 + 2 x + 1

Steps: Replace x with –x

and simplify. If the right side of

the equation stays the same it is an even function.

If every term on the right side changes sign, then the function is odd.

PRACTICE:

Determine if function is Even, Odd or Neither

a. f (x) = x2 + 6b. g(x) = 7x3 – xc. h (x) = x5 + 1

The function on the left is even.

What does that mean in terms of the graph of the function?

The graph is symmetric with respect to the y-axis. For every point (x, y) on the graph, the point (-x, y) is also on the graph.

All even functions have graphs with this kind of symmetry.

The graph of function f (x) = x3 is odd.

It may not be symmetrical with respect to the y-axis. It does have symmetry in another way.

Can you identify how?

For each point (x, y) there is a point (-x, -y) is also on the graph.

Ex. (2, 8) and (-2, -8) are on the graph. The graph is symmetrical with respect to the

origin. All ODD functions have graphs with origin

symmetry.

SUMMARY:ANSWER IN COMPLETE

SENTENCES.

What does it mean if a function f is increasing on an interval?

If you are given a function’s equation, how do you determine if the function is even, odd or neither?

Determine whether each function is even, odd or neither.

a. f (x) = x2- x4 b. f (x) = x(1- x2)1/2

AIM #1.3B: HOW DO WE UNDERSTAND AND USE PIECEWISE FUNCTIONS?

A piecewise function is a function that is defined by two (or more) equations over a specified domain.

Example: ( DO NOT COPY) READ

A cellular phone company offers the following plan:

$20 per month buys 60 minutes Additional time costs $0.40 per minute

HOW DO WE EVALUATE A PIECEWISE FUNCTION?

PRACTICE:

Find and interpret each of the following:a. C (40) b. C (80)

HOW DO WE GRAPH A PIECEWISE FUNCTION?

We can use the graph of a piecewise function to find the range of f.

What would the range be for the piecewise function? ( For previous piecewise function)

Some piecewise functions are called step functions because the graphs form discontinuous steps.

One such function is called the greatest integer function, symbolized by int (x) or

int (x) = greatest integer that is less than or equal to x.

For example:a. int (1) = 1, int (1.3) = 1, int (1.5) = 1, int (1.9)= 1

b. int (2) = 2, , int (2.3) =2 , int (2.5) = 2, int (2.9)= 2

x

Graph of a Step Function

FUNCTION AND DIFFERENCE QUOTIENTS

Definition of the Difference Quotient of a Function:

The expression for h≠0 is called

the difference quotient of the function f.

f (x h) f (x)h

HOW DO WE EVALUATE AND SIMPLIFY A DIFFERENCE

QUOTIENT?

If f (x) = 2x2 – x + 3, find and simplify each expression:

f ( x + h)

Try:

Steps: Replace x with (x +

h) each time x appears in the equation.

f (x h) f (x)h

h 0

PRACTICE:

If f (x)= -2x2 + x + 5, find and simplify each expression:

a. f (x + h)b.

f (x h) f (x)h

h 0

SUMMARY: ANSWER IN COMPLETE

SENTENCES.

What is a piecewise function? Explain how to find the difference quotient of

a function f,

If an equation for f is given.

f (x h) f (x)h

h 0

AIM #1.4 HOW DO WE IDENTIFY A LINEAR FUNCTION AND ITS SLOPE?

Data presented in a visual form as a set of points is called a scatter plot. A scatter plot shows the relationship between two types of data.

Regression line is the line that passes through or near the points. This is the line that best fits the data points.

We can write an equation that models the data and allows us to make predictions.

WHAT IS THE DEFINITION OF A SLOPE OF A LINE?

PRACTICE:

Find the slope of a line.a. (-3, 4) and (-4, -2)b. (4, -2) and (-1, 5)

POINT-SLOPE FORM EQUATION

The point-slope of the equation of a nonvertical line with slope m that passes through the point (x1,y1) is:

y –y1=m (x – x1)

HOW DO WE WRITE AN EQUATION IN POINT-SLOPE FORM?

Write an equation in point-slope form for the line with slope 4 that passes through the point (-1, 3).

Then solve the equation for y.

Steps: Write the point-slope form equation.

y –y1=m (x – x1) Substitute the given values.

Then solve for y.

PRACTICE:

Write an equation in point-slope form for the line with slope 6 that passes through the point (2, -5).

Then solve the equation for y.

HOW DO WE WRITE AN EQUATION FOR A LINE WHEN WE ONLY HAVE TWO POINTS?

Write an equation in point-slope form for the line that passes through the points (4, -3) and (-2, 6).

Then solve the equation for y.What would you need to do first?

Steps: Find the slope. Then choose any pair of points and substitute

into the point-slope form equation. Then solve the equation for y.

PRACTICE:

Write an equation in point-slope form for the line that passes through the points (-2,-1) and (-1, -6).

Then solve the equation for y.

SUMMARY: ANSWER IN COMPLETE

SENTENCES.

What is the slope of a line and how is it found?

Describe how to write an equation of a line if you know at least two points on that line.

How do you derive the slope-intercept equation from y –y1=m (x – x1)?

AIM # 1.4B: HOW DO WE WRITE AND GRAPH LINEAR EQUATIONS IN THE FORM

OF Y = MX+B?

Slope intercept equation is y = mx + b where m is the slope and b is the y-intercept of the equation.

Graphing y = mx + b using the slope and y-intercept.

1. Graph the y-intercept first. (0, b)2. Then use the slope to get to the other points

on the line.Let’s try:

f (x)3

2x 2

EQUATIONS OF HORIZONTAL LINES

A horizontal line has a m=0. Therefore the equation is y=0x + b which can be simplified to

y = b.

Graph y = 3 or f (x) = 3 Note: This is a constant function

EQUATIONS OF VERTICAL LINES

The slope of a vertical line is undefined. The equation of a vertical line is x = a, where a is the x-intercept of the line.

Note: No vertical line represents a function.

WHAT IS THE GENERAL FORM OF THE EQUATION OF A LINE?

Every line has an equation that can be written in the general form:

Ax + By = C or Ax + By- C = 0

Where A, B, C are real numbers and A and B are not both zeros.

FINDING THE SLOPE AND THE Y-INTERCEPT

Find the slope and the y-intercept of the line whose equation is 3x + 2y – 4 =0.

Steps:1. The equation is given in general form.

Express in y = mx + b by solving for y.

1. Then the slope and y-intercept can be identified.

PRACTICE:

Find the slope and the y-intercept of the line whose equation is 3x + 6y – 12 =0.

Then use slope and y-intercept to graph the line.

HOW DO WE FIND THE INTERCEPTS FROM THE GENERAL FORM OF THE EQUATION OF A LINE?

Graph using the intercepts: 4x – 3y – 6=0

Steps: To find the x-intercept. Set y = 0 and solve

for x.

To find the y-intercept. Set x = 0 and solve for y.

Graph the points and draw a line connecting these points.

PRACTICE:

Graph using the intercepts: 3x – 2y – 6=0

REVIEW OF THE VARIOUS EQUATIONS OF LINES

SUMMARY:ANSWER IN COMPLETE

SENTENCES.

How would you graph the equation x = 2. Can this equation be expressed in slope-intercept form? Explain.

Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.

How do you use the intercepts to graph the general form of a line’s equation?

Aim # 1.5: How do we find the average rate of change? Slope as a rate of

change: Example 1:

The line on the graph for the number of women and men living alone are shown in the graph.

Describe what the slope represents.

Solution: Note x – represents the year and y- number of women.

Choose two points from the women’s graph.

Find the slope and then describe the slope.

Remember to include the units.

xinchange

yinchangem

Practice: Use the ordered pairs

and find the rate of change for the green line or men graph. Express slope two decimal places and describe what it represents.

Average Rate of Change

If the graph of the function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points.

This line is called the secant line.

Problem:

Looking at the graph, what is the man’s average growth rate between the ages 13 and 18.

The Average Rate of Change of a Function

Example 1: Find the average rate of

change of f (x) = x2 from Steps: Use

12

12 )()(

xx

xfxf

x

y

10. 21 xtoxa

Find the average rate of change of f (x) = x2 from

02.

21.

21

21

xtoxc

xtoxb

Average Velocity of an Object

Suppose that a function expresses an object’s position, s (t), in terms of time, t. The average velocity of the object from t1 to t2 is:

12

12 )()(

tt

tsts

t

s

Example 2:

The distance, s (t), in feet, traveled by a ball rolling down a ramp is given by the function:

s (t)= 5t2, where t is the time, in seconds after the ball is

released. Find the ball’s average velocity from

a. t1 = 2 seconds to t2 = 3 seconds

b. t1 = 2 seconds to t2 = 2.5 seconds

c. t1 = 2 seconds to t2 = 2.01 seconds

Summary: Answer in complete sentences.

If two lines are parallel, describe the relationship between their slopes and y-intercept.

If two line are perpendicular, describe the relationship between their slopes.

What is the secant line? What is the average rate of change of a

function?

Aim #1.6: How do we recognize transformations?

Review Algebra’s Common Graphs (distribute)

Vertical Shift

Vertical Shifts

In general if c is positive, y = f (x) +c shifts upward c units. If c is negative it shifts downward c units.

Example 1:

Practice:

Use the graph of to obtain the graph of

xxf )(

3)( xxg

Horizontal Shifts

In general, if c is positive, y = f (x + c) shifts the graph of f to the left c units and y = f (x – c) shifts the graphs of f to the right c units.

These are called horizontal shifts of the graph of f.

Example 2:

Note:

Example 3:

Use the graph of f (x ) = x2 to obtain the graph of h (x) = (x + 1)2 – 3.

Steps to combining a shift: Graph the original function. Then shift horizontally. Then shift vertically.

Reflections of Graphs

Reflection about the x-axis:

The graph of y = - f (x) is the graph of y = f

(x) reflected about the x- axis.

Example 4: Use the graph of to obtain the

graph of

3)( xxf

3)( xxg

Reflections of GraphsReflection about the y-axis:

The graph of y = f (-x) is the graph of y = f (x) reflected over the y –axis.

Example: The point (2, 3) reflected over the y-axis is (-2, 3).

Vertical Stretching

Let f be a function and c be a positive real number.

If c > 1 the graph of y = c f (x) is the graph y = f (x) vertically stretched.

How? By multiplying the y-coordinates by c.

Vertical Shrinking Let f be a function and c be a positive real

number. If 0 < c < 1 the graph of y = c f (x) is the

graph y = f (x) vertically shrunk. How?

By multiplying the y-coordinates by c.

Example 6:

Use the graph of f (x) = x3 to obtain the graph of

h (x) =3

2

1x

Horizontal Shrinking Let f be a function and c be a positive real

number. If c > 1 the graph of y = f (cx) is the

graph

y = f (x) horizontal shrink. How?

By dividing x- coordinates by c.

Horizontal Stretching

Let f be a function and c be a positive real number.

If 0 < c < 1 the graph of y = f (cx) is the graph

y = f (x) horizontal stretch. How?

By dividing x- coordinates by c.

Example 7:

Use the graph y = f (x) to obtain each of the following graphs.

a. g (x) = f (2x) b. h( x) = f( 1/2x)

Sequences of Transformations

Transformations involving more than one transformation can be graphed performing the transformations in the following order:

Horizontal shiftingStretching or shrinking

ReflectingVertical shifting

Summary: Answer in complete sentences.

What must be done to a function’s equation so that its graph is shifted vertically upward?

What must be done to a function’s equation so that its graph is shifted horizontally to the right?

What must be done to a function’s equation so that its graph is reflected about the x-axis?

What must be done to a function’s equation so that its graph is stretched vertically?

Aim# 1.7: What are composite functions?

Finding the domain of a function (w/ no graph).

Example 1: Find the domain of each function.

123)(.

32

23)(.

7)(.

2

2

xxhc

xx

xxgb

xxxfa

The Algebra of Functions

We can combine functions by addition, subtraction, multiplication and division by performing operations with the algebraic expressions that appear on the right side of the equations.

Example 2: Let f (x)=2x – 1 and g (x) = x2 + x – 2

Find the following function.Sum : (f + g ) (x)= f (x ) + g (x)

( 2x – 1) + (x2 + x – 2)= 1. Substitute given functions.

2. Simplify.

The Algebra of Functions

Example 3: Adding Functions and Determining the Domain

Find each of the following:a. (f + g) (x)b. The domain of (f + g)

2)(3)( xxgandxxfLet

Composite Functions

This is another way to combine functions. Example: f (g (x)) can be read as f of g of x

and it is written as

))(())(( xgfxgf

Example 4: Forming Composite Functions

Given f (x)= 3x – 4 and g (x) = x2 – 2x + 6, find each of the following:

)1)((.

))((.

))(.(

fgc

xfgb

xgfa

Example 5: Forming a Composite Function and Finding Its Domain

gfofdomaintheb

xgfa

eachfindx

xgandx

xf

.

))((.

:,3

)(1

2)(

Practice:

Given and g(x)=

Find each of the following:1. 2. the domain of

2

4)(

x

xf

x

1

))(( xgf

))(( xgf

Summary: Answer in complete sentences.

Explain how to find the domain of a function of a radical and a rational equation.

If equations for f and g are given, explain how find f- g.

If equations for two functions are given, explain how to obtain the quotient of the two functions and its domain. ( ex. f/g(x))

Describe a procedure for finding . What is the name of this function?

))(( xgf

Aim# 1. 8 What is an inverse function?

Example 1: Verifying Inverse Functions Show that each function is the inverse of the

other:

Steps: To show that f and g are inverses of each

other, we must show that f (g (x)) = x and f (g (x)) = x.

Begin with f (g (x)) Then with g (f (x)) Do you get x?

3

2)(23)(

x

xgandxxf

Practice:

Show that each function is the inverse of the other:

4

7)(74)(

x

xgandxxf

How do we find the inverse? Find the inverse of f (x) = 7x – 5

Steps: Replace f (x) with y.

Switch x and y.

Solve for y.

Then replace y with f-1 (x)

How do we find the inverse? Find the inverse of:

f (x ) = x3 + 1

Steps: Replace f (x) with y.

Switch x and y.

Solve for y.

Then replace y with f-1 (x)

How do we find the inverse? Find the inverse of: Steps:

Replace f (x) with y.

Switch x and y.

Solve for y.

Then replace y with f-1 (x)

45

)( x

xf

Practice:

Find the inverse for the following:

Steps: Replace f (x) with y.

Switch x and y.

Solve for y.

Then replace y with f-1 (x)1

3)(

14)(

72)(3

xxf

xxf

xxf

Properties of Inverse Functions

1. A function f has an inverse that is a function, f-1, if there is no horizontal line that intersects the graph of the function f at more than one point.

Properties of Inverse Functions

2. The graph of a function’s inverse is a reflection of the graph of f about the line y = x.

Summary: Answer in complete sentences.

Explain how to determine if two functions are inverse of each other.

Describe how to find the inverse of a function.

What are some properties of a function and its inverse? Draw an illustration to support your written statement.

Aim #1.9: How do we find the distance and midpoint of a segment? Formulas:

2,

2:int

)()(tan

2121

212

212

yyxxformulaMidpo

yyxxdcedis