Unit 7: Non-Linear Equations & Real-World 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 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

2 3xy

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. Matcha) y = 3x + 5 i) exponential growth

b) ii) exponential decay c) iii) constant increase• Sections of the book to read: 8-2, 8-3, 8-4

4 5x

y 1

45

x

y

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:• 1) find the axis of symmetry• 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)

2

bx

a

• Graph• Ex1. y = 3x²• Ex2. y = 2x² + 6x – 1 • Ex3. y = -x² + 4x + 1• Ex4.

• The larger a becomes, the skinnier the graph

• Sections of the book to read: 9-1 and 9-2

213 1

2y x x

Section 6: The Quadratic Formula• This formula is used in many math and science

classes, MEMORIZE it!• The quadratic formula• 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

2 4

2

b b acx

a

• 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 up-side-down parabola (because it is a quadratic equation)

• If an object is launched, the equation would be

• 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

20 0

1

2h gt v t h

• 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

• 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

216 65 9h t t

• 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

following equation:216 40 2y x x

• 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. Evaluatea) b) c)5 9 2 8 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• Ex3. Solve• Ex4. Solve

6x

3 15x

5 20m

• To find the distance between two points on a number line, called x1 and x2,

• 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, • To find the distance between two points on the

coordinate plane,

2 1x x

2x x

2 2

2 1 2 1x x y y

• 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