1. Non-Linear Functions and Real-World Applications

2. Section 1: Compound Interest Terminology reminders: 5x³ has 5 as the coefficient, x as the base, 3 as the exponent and x³ is the power Simple interest is one way to calculate interest, but it is rarely used Simple interest: I = prt I = interest, p = principal (starting amount), r = rate (interest rate in decimal form), and t = time (given in years) Ex1. Dean deposits $800 in to a bank that uses simple interest. If the interest rate is 5.4% and the money sits for 10 years, how much money will be in the account?

3. Compound is more beneficial and more frequently used It is more beneficial because your money is earning more money and it grows exponentially (unlike the linear earnings of simple interest) Compound interest: T = total amount, P = principal amount (beginning), r = interest rate (given as a percent), and t = time (in years) Notice that you are calculating the total, not the interest Ex2. Fred deposited $800 into a bank that uses compounded interest for 10 years. If the annual interest rate is 5.4%, how much will be in the account? Compare examples 1 and 2 to see how different compound and simple interest are

4. Ex3. George has $2000 to invest in a bank using compound interest. The annual interest rate is. A) How much will be in the account after 3 years? B) How much will be in the account after n years? Section of the book to read: 8-1

5. Section 2: Exponential Growth Compound interest is an example of exponential growth using money The formula for exponential growth was slightly adapted for monetary purposes Exponential growth: b = beginning amount, g = growth factor (you will have to add or subtract from 1 if it is given in percent form), x = 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 in two years?

6. Zero Exponent Property: If g > 1, then it is exponential growth If g = 1, then there is no change If g < 1, then it is exponential decay The graph of exponential growth is called an exponential growth curve (see #1 below) The graph of exponential decay is called an exponential decay curve (see #2 below) #1#2

7. Ex2. Ten frogs are introduced to an area. The population increases by 30% every six months. How many frogs will there be in 5 years? When graphing exponential curves, you must make a table containing at least 4 points Make at least one x-value negative unless it is a real-world situation that can’t be negative Ex3. Graph Section of the book to read: 8-2

8. Section 3: Exponential Decay Exponential decay uses the same exponential formula except that the growth factor is less than 1 Since the amount is being repeatedly multiplied by a number less than one, it decreases rapidly If an exponential decay question is given in terms of a percent lost, you will need to SUBTRACT that percent from 100% and convert to a decimal Ex1. The population of a town is 650,000, but they are losing approximately 4% of the population each year. How many people will be left in the town after 2o years?

9. Ex2. $200,000 was invested into the stock market, but loses about 5.5% per year. How much money will there be in the account after 3 years? Ex3. Match each of the following A) i) exponential growth B) ii) exponential decay C) iii) constant increase

10. Ex4. Graph Sections of the book to read: 8-2, 8-3, 8-4

11. Section 4: Graphing Quadratic Equations A quadratic function is one that can be written in the form of y = ax² + bx + c where a ≠ 0 The degree of all quadratic functions is 2 The graph of a quadratic function is a parabola The line that cuts a parabola in half is called the axis of symmetry (or the line of symmetry) The equation will be x = a number The point where the graph touches the axis of symmetry is the vertex

12. To graph a quadratic function: 1) find the axis of symmetry 2) use the axis of symmetry to find the vertex 3) make a table of at least 5 values including the vertex and two points on either side of the vertex 4) connect the points Using ax² + bx + c If a > 0 (positive), then the parabola opens up and the vertex is the minimum If a < 0 (negative), then the parabola opens down and the vertex is the maximum The larger the absolute value of a, the skinnier the parabola will be

13. Graph each of the following: Ex1.Ex2. Ex3. Sections of the book to read: 9-1 and 9-2

14. Section 5: The Quadratic Formula To find the values of the x-intercepts (the solutions to the equation), use the quadratic formula You will have to memorize the formula The quadratic formula: You will have to use the formula in several small steps to avoid calculator errors You have to use the formula once with the + and then again with the – to get both answers In order to use the quadratic formula, make sure the equation is set = 0

15. Real-world situations might require that you disregard one of the answers obtained The Discriminant is part of the quadratic formula: b² ─ 4ac If it is a perfect square, then the answers will be rational, otherwise they will be irrational Solve and round to the nearest hundredth where necessary Ex1. 5x² + 3x – 6 = 0 Ex2. 2x² ─ 4x = 3 Ex3. Solve and give the exact answers. 6x² + 5x – 8 = 0 Ex4. Without solving, tell whether the answers will be rational or irrational for -x² + 7x = -2 Sections of the book to read: 9-5 and 9-6

16. Section 6: Projectiles The height above ground of an object measured over time after it has been launched or thrown results in a quadratic function (and therefore the graph is a parabola) The equation is given by: g is acceleration due to gravity (32 ft/sec² or 9.8 m/sec²) Initial velocity is Initial height is Open your book to page 567 The vertex will be the maximum height that is reached by the object

17. To find the times when the object reaches a certain height, set the equation equal to the height, subtract that height from both sides, and then solve using the quadratic formula Ex1. The height of a launched object is given by: 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? When graphing these functions, place time on the x-axis and height on the y-axis If the Discriminant is positive, then there are 2 real solutions

18. 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: y = -16x² + 40x – 8 Ex3. Graph the equation from example 1 Sections of the book to read: 9-4 and 9-6

19. Section 7: Absolute Value and Distance The absolute value of a number is its’ distance from 0 on the number line It must be positive because it is a distance Numbers that are opposites have the same absolute value (i.e. 5 and -5 have an absolute value of 5) Absolute value symbols work like other grouping symbols, do what is inside of them first and then find the absolute value Ex1. Evaluate a) b)c)

20. The graph of an absolute value function is v-shaped To graph an absolute value function, you need to make a table of values At least the vertex and 2 points on either side Absolute value functions have two answers because two numbers have the same absolute value Solve Ex2. Ex3. Ex4. To find the distance between two points on a number line: Ex5. Find the distance between -31 and 24 on the number line

21. For all values of x, To find the distance between two points on the coordinate plane: Ex6. Find the exact distance between A = (-4, 6) and B = (6, 2) Sections of the book to read: 2-5, 9-8, 9-9, and 13-3