Download Unit 7: More With Exponents & Real

Unit 7: Non-Linear Equations & RealWorld Applications
Section 1: Compound Interest
• Using 5x³: 5 is the coefficient, x is the base, 3 is
the exponent and x³ is the power
• There is more than one way to calculate interest
• The simple interest formula: I = prt
I = interest, p = principal, r = rate (in decimal
form) and t = time (in years)
• This formula is not used very often
• Ex1. Lee deposits $500 in a bank with an interest
rate of 6%. Using simple interest, how much will be
in the account after 4 years?
• Compound Interest is more commonly used by
banks and other lending institutions
• With compound interest, the interest you earn
then earns more interest (your money grows
faster)
• The compound interest formula gives you the
total, not just the interest
• Compound Interest: T = P(1 + i)n
T = total, P = principal, i = interest rate (as a decimal),
and n = number of years
• Ex2. Suppose you deposit P dollars in a savings
account upon which the bank pays an annual yield
of 4%. If you make no other deposits or
withdrawals, how much money will be in the
account at the end of 1 year?
• Ex3. Fred deposited $500 in a bank with an interest
rate of 6%. Using the compound interest formula,
how much will be in the account after 4 years?
• Compare your answers from Ex1. and Ex3.
• Ex4. Use the information from Ex3. but change the
annual yield to 5½%.
• Section from the book to read: 8-1
Section 2: Exponential Growth
• Compound interest is a real-world example of
exponential growth (the amount grows
exponentially)
• Exponential growth: y = b · gx where b is the
beginning amount (initial amount), g is the
growth factor, and x is the number of times the
growing occurs
• Ex1. Twenty-five rabbits are introduced to an
area. Assume that the population triples every
four months, how many rabbits will there be after
3 years?
• You can use exponential growth to demonstrate
that anything to the zero power is equal to one
(Zero Exponent Property)
• For an exponential function to be growth (the
number gets larger), the growth factor g must be
greater than 1
• If g = 1, then there is no change
• If g < 1, then it is exponential decay
• The growth factor in compound interest = 1 + i
• The graph of exponential growth is known as an
exponential growth curve
• Open your book to page 494, we are going to look
at the exponential growth curve in example 2
• The graph begins looking flat, but quickly
increases
• To graph any exponential function, make a table
of values and graph the points until you have
enough to make a distinguishable curve
• Ex2. Ten frogs are introduced into an area. The
population is increasing by approximately 30% per
year. How many frogs will there be in 5 years?
• Ex3. Graph y  2  3
for x = -2, -1, 0, 1, 2
• We will study
exponential decay
and exponential
decay curves later
in this unit
• Section of the book
to read: 8-2
x
Section 3: Comparing Exponential
Growth and Constant Increase
• With both constant increase and exponential
growth, the amount is increasing
• The graph of constant increase is a line with a
positive slope (and that slope remains constant)
• The graph of exponential growth is a curve with
no constant slope (it is different between each
pair of points)
• With constant increase, a number is repeatedly
added to the total
• With exponential growth, a number is repeatedly
multiplied by the total
• In a real-world example of exponential growth (or
constant increase), a Quadrant I only graph is all
you need
• Make the increment on your axes clear and
appropriate
• Open your book to page 500, we are going to
look at example 2
• Section of the book to read: 8-3
Section 4: Exponential Decay
• Exponential decay uses the same formula as
exponential growth y = b · gx , but now the growth
factor must be less than one (g < 1)
• With exponential decay, the total amount is
decreasing (repeatedly being multiplied by a
number less than 1)
• The graph of exponential decay is the reflection of
exponential growth (see page 507)
• The graph decreases rapidly, but then starts to
level off
• Ex1. The population of a town begins at 650,000,
but they are losing approximately 4% of their
population each year. How many people will be
left in the town after 20 years?
• Add the percent (as a decimal) to 1 when it is
increase and subtract it from 1 for decrease
• Ex2. Match
a) y = 3x + 5
i) exponential growth
x
b) y  4  5
ii) exponential decay
c) y  4  1 
iii) constant increase
5
• Sections of the book to read: 8-2, 8-3, 8-4
x
Section 5: Graphing Quadratic Equations
• A quadratic function is one that can be written in
the form y = ax² + bx + c where a ≠0
• A quadratic function must have a degree of 2
• The graph of a quadratic function is a parabola
(see page 548)
• If b and c are both = 0, then the parabola will
have reflectional symmetry over the y-axis
(choose x = -2, -1, 0, 1, 2 for your table)
• To graph a parabola, make a table of values and
place them on the graph
• You only need to find the vertex and two points
on either side of the vertex to be able to graph
the shape accurately enough
• To find the vertex:
b
• 1) find the axis of symmetry x  2a
• 2) use that x-value as the x-value of the point,
plug it into the equation to find the
corresponding y-value
• If a > 0, then the parabola opens up (has a
minimum)
• If a < 0, then the parabola opens down (has a
maximum)
•
•
•
•
•
Graph
Ex1. y = 3x²
Ex2. y = 2x² + 6x – 1
Ex3. y = -x² + 4x + 1
Ex4. y  1 x 2  3x  1
2
• The larger a
becomes, the
skinnier the graph
• Sections of the
book to read: 9-1
and 9-2
Section 6: The Quadratic Formula
• This formula is used in many math and science
classes, MEMORIZE it!
b  b2  4ac
• The quadratic formula x 
2a
• The quadratic formula allows you to find where
the graph crosses the x-axis (these are the
solutions to the equation)
• You have to go through the formula twice to get
both potential answers
• Quadratic equations must be = 0 to use the
formula
• In a real-world scenario, one of the solutions may
be disallowed because it does not make sense
• Solve. Round to the nearest hundredth if
necessary.
• Ex1. 5x² + 3x – 6 = 0
• Ex2. 2x² – 4x = 3
• Ex3. Solve. Give the exact answers
6x² + 5x – 8 = 0
• b² - 4ac is known as the discriminant
• If the discriminant is a perfect square, then the
answer is rational, otherwise it is irrational
• Ex4. Without solving, tell whether the answers
will be rational or irrational. -x²+ 7x = -2
• Be VERY careful when using your calculator and
this formula!!! Do one step at a time!!!
• Sections of the book to read: 9-5 and 9-6
Section 7: Projectiles
• If you measure the height of a projectile over time
(after it is launched) the graph would be an upside-down parabola (because it is a quadratic
equation)
• If an object is launched, the equation would be
1 2
h
gt  v0t  h0
2
• g is a set number (it stands for acceleration due to
gravity): 32 ft/sec or 19.6 m/sec
• v0 is the initial velocity
• h0 is the initial height
• Open your book to page 567
• The maximum height will be the vertex, so we
can use the algorithm from section 5
• Ex1. An object launched is given the equation
h  16t  65t  9
2
• A) What is the initial height of the object?
• B) When will the object hit the ground?
• C) When will the object be 50 feet above
ground?
• Time is to be graphed on the x-axis, height on the
y-axis
• The discriminant is b² - 4ac
• If the discriminant is positive, then there are two
real solutions (although 1 may not be reasonable)
• If the discriminant is negative, then there are no
real solutions
• If the discriminant is 0, then there is one real
solution
• Ex2. How many real solutions will there be to the
2
following equation: y  16 x  40 x  2
• Ex3. Graph the
equation from
Ex2.
• Sections of the
book to read: 9-4
and 9-6
Section 8: Absolute Value and Distance
• The absolute value of a number is the distance
that number is from zero on a number line
• The absolute value of a number is always positive
because it is a distance
• Opposite numbers have the same absolute value
• Treat absolute value symbols like other grouping
symbols (simplify within and then find the
absolute value)
• Ex1. Evaluate
a) 5  9
b) 2  8
c) 6  9  3  6
• When you graph an absolute value function, it is
v-shaped with an axis of symmetry (it has
reflectional symmetry over a vertical line)
• To graph an absolute value function, make a table
of values and look for symmetry (you need a
minimum of the vertex and two numbers on
either side)
• Ex2. Solve x  6
• Ex3. Solve x  3  15
• Ex4. Solve m  5  20
• To find the distance between two points on a
number line, called x1 and x2, x2  x1
• Ex5. Find the distance between -12 and 9 on the
number line
• Before we discussed the idea that taking the
square root of a square results in the initial
number, but that is not completely true
• For all values of x, x2  x
• To find the distance between two points on the
coordinate plane, x  x 2  y  y 2
2
1
2
1
• If the line is vertical or horizontal, you can use the
formula for finding distance on a number line
because either the x-values or the y-values are
the same
• Ex6. Find the distance between A = (-3, 5) and B =
(8, 12) on the coordinate plane (exact form)
• Sections of the book to read: 2-5, 9-8, 9-9, 13-3