Concavity
The graph at the right
represents a portion of
a function f over the
interval x1, x2.
y
o
f
x1
x2
x
Over the mentioned interval, the graph is increasing and bending down when
we analyze it from left to right. Left to right is standard direction
Quadratic Functions
Quadratic functions have the following structure.
f ( x ) = ax 2 + bx + c
The formula above is known as the standard form of quadratic functions.
The characteristic term is the one including x2. The other terms could be missing
but x2
Graphs of Exponential
Functions
We have already discussed the graph of exponential functions in lesson 3.1.
In this lesson, we will discuss the effect of a and b in the equation y = abx.
Effect of the parameter a
The figure below shows the graphs of y = 5
THE SINE AND COSINE
FUNCTION
In lesson 6.1, we obtained the graph below.
y
40
o
x
8
16
That shape is new to us. We do not have a formula to model that kind of graph yet.
Because we obtained that shape by analyzing the height of a passenger on a ferris
whe
Continuous Growth and the
number e
Example 1
Lets consider a savings account where the initial deposit is $1 and the
annual interest rate is 100%.
Find the balance in the account when t = 1 year and interest rate is
computed
a) Annually
e) Daily
b) Semi a
Comparing Exponential and
Linear functions
Lets analyze how y changes when x increases by 1, for linear functions
For linear function y = 2 + 3x:
If x = 0 y = 2
If x = 1 y = 2 + 3
If x = 2 y = 2 + 3 + 3
If x = 3 y = 2 + 3 + 3 + 3
For linear function y = 5
How did mathematicians from the 17th century solve the
following equation?
2 =10
t
The equation above is called Exponential Equation
because the unknown is an exponent.
To solve that equation, mathematicians used a method
involving trial and error.
Before
Compound Interest
In the note before Example 2, from lesson 3.4, we concluded that the balance
when the initial amount is A and the annual interest rate is r can be computed
by using
r
B (t ) = A 1 +
n
nt
where n = number of times interest is computed
INVERSE TRIGONOMETRIC
FUNCTIONS
When computing the sin(40o) the angle 40o is known and the corresponding sine
has to be found.
To find sin(40o) a calculator will use the function y = sin(x), whose outputs are
already built in device, to find that sin(40o)
Exponential Functions
So far, we have discussed the following types of functions.
Type of Function
Typical Equation
Shape of Graph
Linear
y = mx + b
Line
Quadratic
y = ax2 + bx + c
Parabola
Absolute Value
v = |x|
V-shape
The next example will allow us to
Exponential Functions
So far, we have discussed the following types of functions.
Type of Function
Typical Equation
Shape of Graph
Linear
y = mx + b
Line
Quadratic
y = ax2 + bx + c
Parabola
Absolute Value
v = |x|
V-shape
The next example will allow us to
Great exam! Keep up the awesome work!
Problem 1a has a minor error. There is an error in the ordered pair notation for x=4 (-1). Please note
that the ordered pair should read (4,0). The answer you have provided reads like a y-intercept- oops!
Problem 1e i
HW set 4.
1.
Forthepiecewisefunction,findthespecifiedfunctionvalue.
Tofindthevalueoff(x)whenx=8,wehavetousef(x)=6xbecause8 1
F(8)=6 (8)=48
2.
The function is decreasing on (- ,1 and increasing on (-1, ).
3.
Solve.
Bobwantstofenceinarectangulargardeninhisy
GRADE 98/100
COMMENTS: Great exam! Keep up the good work!
Problem 1e has a minor error. The graph you have provided is beautiful, but you have forgotten to label the
equation of the circle on your sketch (-1).
Problem 3 has a calculation error in the last
HW set 2.
1) Write a slope-intercept equation for a line with the given
characteristics.
Passes through (2, -7) and (6, -2)
y 2 y 1
m= x 2x 1 =
5
2(7)
= 4
62
y=mx+b
5
y= 4 x+b. Given point (2,-7)
5
-7= 4 2 +b
b=
19
2
5
Solution: y= 4 x
19
2
2) Write equat
HW set 6
Answer the question.
How can the graph of f(x) = - x+7
x
be obtained from the graph of f(x)=
First, lets compare f(x) = - x+7 to the basic graph
y= f(x)= x
f(x)= x is a reflection of f(x)= x across x-axis.
The given function f(x) = x+7 .
The grap
VERTICAL AND
HORIZONTAL SHIFTS
Vertical shift
Lets say that we have the following graph f(x).
y
10
x
f(x)
Lets try to graph g(x) = f (x) + 5.
10
y
10
x
10
f(x)
g(x) = f (x) + 5.
Lets find
g( 6)
g( 6) = f( 6) + 5
From graph, f( 6) = 0
Then, g( 6) = 0 + 5 =
REFLECTIONS AND
SYMMETRY
Reflections
Lets say that we have the following graph f(x).
y
10
x
f(x)
Lets graph
g(x) = f (x) .
10
y
10
x
10
f(x)
g(x) = f (x)
Lets find
g( 6)
g( 6) = f( 6)
From graph, f( 6) = 0
Then, g( 6) = 0
Point to graph is ( 6, 0)
Point d
HORIZONTAL STRETCHES
AND COMPRESSIONS
Lets say that we have the following graph f(x).
y
10
x
f(x)
Lets graph
g(x) = f (2x) .
10
y
10
x
10
f(x)
g(x) = f (2x)
Lets find
g( 3)
g( 3) = f ( 2( 3) )
= f ( 6)
From graph, f( 6) = 0
Then, g( 3) = 0
Point to graph
VERTICAL STRETCHES
AND COMPRESSIONS
Lets say that we have the following graph f(x).
y
10
x
f(x)
Lets graph
g(x) = 2f (x) .
10
y
10
x
10
g(x) = 2 f (x)
f(x)
Lets find
g( 6)
g( 6) = 2f( 6)
From graph, f( 6) = 0
Then, g( 6) = 0
Point to graph is ( 6 , 0)
Poi
VERTICAL AND
HORIZONTAL SHIFTS
Vertical shift
Lets say that we have the following graph f(x).
y
10
x
f(x)
Lets try to graph g(x) = f (x) + 5.
10
y
10
x
10
f(x)
g(x) = f (x) + 5.
Lets find
g( 6)
g( 6) = f( 6) + 5
From graph, f( 6) = 0
Then, g( 6) = 0 + 5 =
THE FAMILY OF
QUADRATIC FUNCTIONS
In lesson 2.6, we had a first discussion about quadratic functions.
The following are important aspects of that discussion.
The equation of a quadratic function is
f ( x ) = ax 2 + bx + c.
Is graph is the typical parabola
INTRODUCTION TO
PERIODIC FUNCTIONS
So far, we have discussed the following types of functions.
Type of Function
Linear
Quadratic
Typical Equation
Logarithmic
y
y = mx + b
x
y
2
y = ax + bx + c
Absolute Value
Exponential
Shape of Graph
y
y = |x|
y = ab
t
k
Composite and Inverse
Functions
Lets say a stores manager advertises that all prices in her store will be reduced by
10%.
Then, if the price of any item in the store is represented by x, the discounted prices
can be represented by the function f(x) = 0.9x
Piecewise Functions
A piecewise function is a function defined by several functions on different parts
of its domain.
y
9
B
C
6
f
3
6
4
2
o
2
4
6
x
3
A
6
9
D
For instance, the graph shown above is a piecewise function because it is
defined by three f