Chapter on Functions and Graphs.ppt

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. Mapping
2. 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 = 6
2. 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) = 2x
b. 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?
}
2
4
{ 

 x
x

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:
 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 + 6
b. g(x) = 7x3 – x
c. 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:
 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
 
 

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:
 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:
with slope 6 that passes through the point (2, -5).

HOW DO WE WRITE AN EQUATION FOR
A LINE WHEN WE ONLY HAVE TWO
POINTS?
that passes through the points (4, -3) and (-2, 6).
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.

PRACTICE:
that passes through the points (-2,-1) and (-1, -6).

SUMMARY:
 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
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.
 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:
 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.
x
in
change
y
in
change
m 

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
1
2
1
2 )
(
)
(
x
x
x
f
x
f
x
y





1
0
. 2
1 
 x
to
x
a

 Find the average rate of change of f (x) = x2 from
0
2
.
2
1
.
2
1
2
1





x
to
x
c
x
to
x
b

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:
1
2
1
2 )
(
)
(
t
t
t
s
t
s
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.

Practice:
 Use the graph of to obtain the graph of
x
x
f 
)
(
3
)
( 
 x
x
g

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 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
)
( x
x
f 
3
)
( x
x
g 


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
 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
1
x

Horizontal Shrinking
 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
 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:
 What must be done to a function’s equation so that
its graph is shifted vertically upward?
its graph is shifted horizontally to the right?
its graph is reflected about the x-axis?
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.
12
3
)
(
.
3
2
2
3
)
(
.
7
)
(
.
2
2








x
x
h
c
x
x
x
x
g
b
x
x
x
f
a

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.

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
)
( 

 x
x
g
and
x
x
f
Let

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
))
(
(
)
)(
( x
g
f
x
g
f 


Example 4: Forming Composite Functions
 Given f (x)= 3x – 4 and g (x) = x2 – 2x + 6, find each
of the following:
)
1
)(
(
.
)
)(
(
.
)
)(
.(
f
g
c
x
f
g
b
x
g
f
a




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

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

Summary:
 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?
)
)(
( x
g
f 

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
)
(
2
3
)
(




x
x
g
and
x
x
f

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

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)

 Find the inverse of:
f (x ) = x3 + 1
 Steps:
 Switch x and y.
 Solve for y.

 Find the inverse of:  Steps:
 Switch x and y.
 Solve for y.
4
5
)
( 

x
x
f

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

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:
 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
2
1
2
1
2
1
2
2
1
2
y
y
x
x
formula
Midpo
y
y
x
x
d
ce
dis

Chapter on Functions and Graphs.ppt

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