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?


1.
2.
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.
Mapping
Vertical Line Test

DETERMINE WHETHER THE
RELATION IS A FUNCTION a.
b.
{(1, 6), (2, 6), (3, 8), (4, 9)}
{(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 =0.13x
2 -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?
 x 2 + 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
 x 2 + y 2 =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:

1.
2.
Solve each equation for y and then determine whether the equation defines y as a function of x.
2x + y = 6 x 2 + y 2 = 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) = 0.13x
2 -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) = x 2 + 3x + 5


Substitute the 2 for x and evaluate.
Then repeat.



Evaluate each of the following: f (2) f (x + 3)
 f (-x)

GRAPHS OF FUNCTIONS

 a.
b.
The graph of a function is the graph of the ordered pairs.
Let’s graph: f (x) = 2x 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:
{ x

4
 x

2 }

Using Interval
Notation: [-4, 2]

What would it look like for the range using both set builder and interval notation?

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 (x
1 x
1
<x
) < f(x
2
2
) whenever for any x
1 and x in the interval.
2


A function is decreasing on an open interval, I, if f(x
1
) > f
(x
2
) whenever x
1
> x
2 for any x
1 and x the interval.
2 in


A function is constant on an open interval, I, f(x
1
) = f (x and x
2
2
) for any x
1 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) = x 3 - 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) = x 4 - 2x 2
 h(x) = x 2 + 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:
 a.
b.
c.
Determine if function is Even, Odd or Neither f (x) = x 2 + 6 g(x) = 7x 3 – x h (x) = x 5 + 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) = x 3 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) = x 2 - x 4 b. f (x) = x(1- x 2 ) 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.


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

Graph of a Step Function

FUNCTION AND DIFFERENCE
QUOTIENTS

Definition of the Difference Quotient of a
Function: h the difference quotient of the function f.



HOW DO WE EVALUATE AND
SIMPLIFY A DIFFERENCE
QUOTIENT?


If f (x) = 2x 2 – x + 3, find and simplify each expression: f ( x + h)

Steps:

Replace x with (x + h) each time x appears in the equation.

Try: f ( x  h )  f ( x ) h  0 h


PRACTICE:
 a.
If f (x)= -2x 2 + x + 5, find and simplify each expression: f (x + h) b.
f ( x  h )  f ( x ) h  0 h

SUMMARY:
ANSWER IN COMPLETE
SENTENCES.

What is a piecewise function?

Explain how to find the difference quotient of a f ( x  h )  f ( x ) h  0 h

If an equation for f is given.


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.
b.
(-3, 4) and (-4, -2)
(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
(x
1
,y
1
) is: y –y
1
=m (x – x
1
)

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 –y
1
=m (x – x
1
)
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 –y
1
=m (x – x
1
)?

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.

1.
Graphing y = mx + b using the slope and yintercept.

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
2 x  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.

1.
Steps:
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 ynumber of women.
Choose two points from the women’s graph.
Find the slope and then describe the slope.


Remember to include the units.

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) = x 2 from a .
x
1

0 to x
2

1


Steps:
Use
 y
 x
 f ( x
2
)
 x
2
 f x
1
( x
1
)


Find the average rate of change of f (x) = x 2 from b .
x
1

1 c .
x
1
 
2 to to x
2 x
2

2

0

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 t
1 to t
2 is:
 s
 t
 s ( t
2
) t
2
 s ( t
1
)
 t
1

Example 2:
 a.
The distance, s (t), in feet, traveled by a ball rolling down a ramp is given by the function: t
1
= 2 seconds to t
2 s (t)= 5t 2 , where t is the time, in seconds after the ball is released. Find the ball’s average velocity from
= 3 seconds b.
c.
t
1 t
1
= 2 seconds to t
2
= 2 seconds to t
2
= 2.5 seconds
= 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:
 of g ( x )
 x

3

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 ) = x 2 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 xaxis.

Example 4:
 g ( x )
 
3 x

Reflections of Graphs

Reflection 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) = x 3 to obtain the graph of h (x) =
1
2 x
3

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 shifting
Stretching or shrinking
Reflecting
Vertical 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.
a b c .
.
.
g h f ( x )

(
( x x
)
)

 x
2 
7 x x
2
3 x

2

2 x

3
3 x

12

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) = x 2 + x – 2
Find the following function.
Sum : (f + g ) (x)= f (x ) + g (x)
( 2x – 1) + (x 2 + x – 2)= 1. Substitute given functions.
2. Simplify.

The Algebra of Functions

Example 3:
Adding Functions and Determining the Domain
Let f ( x )
 x

3 and g ( x ) x

2
 a.
b.
Find each of the following:
(f + g) (x)
The domain of (f + g)

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
( f  g )( x )
 f ( g ( x ))

Example 4: Forming Composite Functions

Given f (x)= 3x – 4 and g (x) = x 2 – 2x + 6, find each of the following: a .( f  g )( x ) b .
( g  f )( x ) c .
( g  f )( 1 )

Example 5: F orming a Composite Function and Finding Its Domain a f
.
(
( x ) f

2 x

1
 g )( x ) and g ( x )

3 x
, find each : b .
the domain of f  g

Practice:

Given f ( x )
 x
4

2 and g(x)= 1

Find each of the following:
1.
( f  g )( x )
2. the domain of
( f  g )( x ) x

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 ( f  g )( x ) .
What is the name of this function?

Aim# 1. 8 What is an inverse function?

Example 1: Verifying Inverse Functions

Show that each function is the inverse of the other: f ( x )

3 x

2 and g ( x )
 x

3
2





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?

Practice:

Show that each function is the inverse of the other: f ( x )

4 x

7 and g ( x )
 x

7
4

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 ) = x 3 + 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.
f ( x )

5 x

4

Switch x and y.

Solve for y.

Then replace y with f -1 (x)

Practice:
Find the inverse for the following: f ( x )

2 x

7 f ( x )

4 x
3 
1 f ( x )

3 x

1


Steps:
Replace f (x) with y.

Switch x and y.

Solve for y.

Then replace y with f -1 (x)

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: dis tan ce
Midpo int
 d

( x
2
 x
1
)
2 
( y
2
 y
1
)
2 formula : x
1
 x
2 ,
2 y
1
 y
2
2